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Simplificación de expresiones/ Descomposición en fracciones simples/ sin(x)/(((1-tan(x/2))/(1+tan(x/2)))*(1+sin(x)))

¿Cómo vas a descomponer esta sin(x)/(((1-tan(x/2))/(1+tan(x/2)))*(1+sin(x))) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
         sin(x)        
-----------------------
       /x\             
1 - tan|-|             
       \2/             
----------*(1 + sin(x))
       /x\             
1 + tan|-|             
       \2/
sin⁡(x)1−tan⁡(x2)tan⁡(x2)+1(sin⁡(x)+1)\frac{\sin{\left(x \right)}}{\frac{1 - \tan{\left(\frac{x}{2} \right)}}{\tan{\left(\frac{x}{2} \right)} + 1} \left(\sin{\left(x \right)} + 1\right)} 
sin(x)/((((1 - tan(x/2))/(1 + tan(x/2)))*(1 + sin(x))))
Simplificación general [src] 
   /       /x\\           
  -|1 + tan|-||*sin(x)    
   \       \2//           
--------------------------
             /        /x\\
(1 + sin(x))*|-1 + tan|-||
             \        \2//
−(tan⁡(x2)+1)sin⁡(x)(sin⁡(x)+1)(tan⁡(x2)−1)- \frac{\left(\tan{\left(\frac{x}{2} \right)} + 1\right) \sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\tan{\left(\frac{x}{2} \right)} - 1\right)} 
-(1 + tan(x/2))*sin(x)/((1 + sin(x))*(-1 + tan(x/2)))
Respuesta numérica [src] 
(1.0 + tan(x/2))*sin(x)/((1.0 - tan(x/2))*(1.0 + sin(x)))
(1.0 + tan(x/2))*sin(x)/((1.0 - tan(x/2))*(1.0 + sin(x)))
Denominador racional [src] 
   /       /x\\          
   |1 + tan|-||*sin(x)   
   \       \2//          
-------------------------
/       /x\\             
|1 - tan|-||*(1 + sin(x))
\       \2//
(tan⁡(x2)+1)sin⁡(x)(1−tan⁡(x2))(sin⁡(x)+1)\frac{\left(\tan{\left(\frac{x}{2} \right)} + 1\right) \sin{\left(x \right)}}{\left(1 - \tan{\left(\frac{x}{2} \right)}\right) \left(\sin{\left(x \right)} + 1\right)} 
(1 + tan(x/2))*sin(x)/((1 - tan(x/2))*(1 + sin(x)))
Denominador común [src] 
                   /x\                   
            1 - tan|-| + 2*sin(x)        
                   \2/                   
-1 - ------------------------------------
                             /x\      /x\
     -1 - sin(x) + sin(x)*tan|-| + tan|-|
                             \2/      \2/
−2sin⁡(x)−tan⁡(x2)+1sin⁡(x)tan⁡(x2)−sin⁡(x)+tan⁡(x2)−1−1- \frac{2 \sin{\left(x \right)} - \tan{\left(\frac{x}{2} \right)} + 1}{\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)} - \sin{\left(x \right)} + \tan{\left(\frac{x}{2} \right)} - 1} - 1 
-1 - (1 - tan(x/2) + 2*sin(x))/(-1 - sin(x) + sin(x)*tan(x/2) + tan(x/2))
Potencias [src] 
      /      /   I*x    -I*x \\                     
      |      |   ---    -----||                     
      |      |    2       2  ||                     
      |    I*\- e    + e     /| /   -I*x    I*x\    
   -I*|1 + -------------------|*\- e     + e   /    
      |        I*x    -I*x    |                     
      |        ---    -----   |                     
      |         2       2     |                     
      \       e    + e        /                     
----------------------------------------------------
                           /      /   I*x    -I*x \\
                           |      |   ---    -----||
  /      /   -I*x    I*x\\ |      |    2       2  ||
  |    I*\- e     + e   /| |    I*\- e    + e     /|
2*|1 - ------------------|*|1 - -------------------|
  \            2         / |        I*x    -I*x    |
                           |        ---    -----   |
                           |         2       2     |
                           \       e    + e        /
−i(i(−eix2+e−ix2)eix2+e−ix2+1)(eix−e−ix)2(−i(eix−e−ix)2+1)(−i(−eix2+e−ix2)eix2+e−ix2+1)- \frac{i \left(\frac{i \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)}{e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}} + 1\right) \left(e^{i x} - e^{- i x}\right)}{2 \left(- \frac{i \left(e^{i x} - e^{- i x}\right)}{2} + 1\right) \left(- \frac{i \left(- e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}\right)}{e^{\frac{i x}{2}} + e^{- \frac{i x}{2}}} + 1\right)} 
   /       /x\\          
   |1 + tan|-||*sin(x)   
   \       \2//          
-------------------------
/       /x\\             
|1 - tan|-||*(1 + sin(x))
\       \2//
(tan⁡(x2)+1)sin⁡(x)(1−tan⁡(x2))(sin⁡(x)+1)\frac{\left(\tan{\left(\frac{x}{2} \right)} + 1\right) \sin{\left(x \right)}}{\left(1 - \tan{\left(\frac{x}{2} \right)}\right) \left(\sin{\left(x \right)} + 1\right)} 
(1 + tan(x/2))*sin(x)/((1 - tan(x/2))*(1 + sin(x)))
Unión de expresiones racionales [src] 
   /       /x\\          
   |1 + tan|-||*sin(x)   
   \       \2//          
-------------------------
/       /x\\             
|1 - tan|-||*(1 + sin(x))
\       \2//
(tan⁡(x2)+1)sin⁡(x)(1−tan⁡(x2))(sin⁡(x)+1)\frac{\left(\tan{\left(\frac{x}{2} \right)} + 1\right) \sin{\left(x \right)}}{\left(1 - \tan{\left(\frac{x}{2} \right)}\right) \left(\sin{\left(x \right)} + 1\right)} 
(1 + tan(x/2))*sin(x)/((1 - tan(x/2))*(1 + sin(x)))
Combinatoria [src] 
   /       /x\\           
  -|1 + tan|-||*sin(x)    
   \       \2//           
--------------------------
             /        /x\\
(1 + sin(x))*|-1 + tan|-||
             \        \2//
−(tan⁡(x2)+1)sin⁡(x)(sin⁡(x)+1)(tan⁡(x2)−1)- \frac{\left(\tan{\left(\frac{x}{2} \right)} + 1\right) \sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\tan{\left(\frac{x}{2} \right)} - 1\right)} 
-(1 + tan(x/2))*sin(x)/((1 + sin(x))*(-1 + tan(x/2)))
Compilar la expresión [src] 
   /       /x\\          
   |1 + tan|-||*sin(x)   
   \       \2//          
-------------------------
/       /x\\             
|1 - tan|-||*(1 + sin(x))
\       \2//
(tan⁡(x2)+1)sin⁡(x)(1−tan⁡(x2))(sin⁡(x)+1)\frac{\left(\tan{\left(\frac{x}{2} \right)} + 1\right) \sin{\left(x \right)}}{\left(1 - \tan{\left(\frac{x}{2} \right)}\right) \left(\sin{\left(x \right)} + 1\right)} 
(1 + tan(x/2))*sin(x)/((1 - tan(x/2))*(1 + sin(x)))
Abrimos la expresión [src] 
                       sin(x)                       
----------------------------------------------------
                               /x\               /x\
                            tan|-|     sin(x)*tan|-|
    1          sin(x)          \2/               \2/
---------- + ---------- - ---------- - -------------
       /x\          /x\          /x\            /x\ 
1 + tan|-|   1 + tan|-|   1 + tan|-|     1 + tan|-| 
       \2/          \2/          \2/            \2/
sin⁡(x)−sin⁡(x)tan⁡(x2)tan⁡(x2)+1+sin⁡(x)tan⁡(x2)+1−tan⁡(x2)tan⁡(x2)+1+1tan⁡(x2)+1\frac{\sin{\left(x \right)}}{- \frac{\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}}{\tan{\left(\frac{x}{2} \right)} + 1} + \frac{\sin{\left(x \right)}}{\tan{\left(\frac{x}{2} \right)} + 1} - \frac{\tan{\left(\frac{x}{2} \right)}}{\tan{\left(\frac{x}{2} \right)} + 1} + \frac{1}{\tan{\left(\frac{x}{2} \right)} + 1}} 
   /       /x\\          
   |1 + tan|-||*sin(x)   
   \       \2//          
-------------------------
/       /x\\             
|1 - tan|-||*(1 + sin(x))
\       \2//
(tan⁡(x2)+1)sin⁡(x)(1−tan⁡(x2))(sin⁡(x)+1)\frac{\left(\tan{\left(\frac{x}{2} \right)} + 1\right) \sin{\left(x \right)}}{\left(1 - \tan{\left(\frac{x}{2} \right)}\right) \left(\sin{\left(x \right)} + 1\right)} 
(1 + tan(x/2))*sin(x)/((1 - tan(x/2))*(1 + sin(x)))
Parte trigonométrica [src] 
   /         2/x\\          
   |    2*sin |-||          
   |          \2/|          
   |1 + ---------|*sin(x)   
   \      sin(x) /          
----------------------------
/         2/x\\             
|    2*sin |-||             
|          \2/|             
|1 - ---------|*(1 + sin(x))
\      sin(x) /
(2sin⁡2(x2)sin⁡(x)+1)sin⁡(x)(−2sin⁡2(x2)sin⁡(x)+1)(sin⁡(x)+1)\frac{\left(\frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} + 1\right) \sin{\left(x \right)}}{\left(- \frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} + 1\right) \left(\sin{\left(x \right)} + 1\right)} 
                    2/x\ /       /x\\               
               8*tan |-|*|1 + tan|-||               
                     \4/ \       \2//               
----------------------------------------------------
                            /           /x\ \       
             2              |      2*tan|-| |       
/       2/x\\  /       /x\\ |           \2/ |    /x\
|1 + tan |-|| *|1 - tan|-||*|1 + -----------|*tan|-|
\        \4//  \       \2// |           2/x\|    \2/
                            |    1 + tan |-||       
                            \            \2//
8(tan⁡(x2)+1)tan⁡2(x4)(1+2tan⁡(x2)tan⁡2(x2)+1)(1−tan⁡(x2))(tan⁡2(x4)+1)2tan⁡(x2)\frac{8 \left(\tan{\left(\frac{x}{2} \right)} + 1\right) \tan^{2}{\left(\frac{x}{4} \right)}}{\left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(1 - \tan{\left(\frac{x}{2} \right)}\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \tan{\left(\frac{x}{2} \right)}} 
             /       /x\\    /x\            
           2*|1 + tan|-||*tan|-|            
             \       \2//    \2/            
--------------------------------------------
                           /           /x\ \
                           |      2*tan|-| |
/       2/x\\ /       /x\\ |           \2/ |
|1 + tan |-||*|1 - tan|-||*|1 + -----------|
\        \2// \       \2// |           2/x\|
                           |    1 + tan |-||
                           \            \2//
2(tan⁡(x2)+1)tan⁡(x2)(1+2tan⁡(x2)tan⁡2(x2)+1)(1−tan⁡(x2))(tan⁡2(x2)+1)\frac{2 \left(\tan{\left(\frac{x}{2} \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}}{\left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(1 - \tan{\left(\frac{x}{2} \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
   /       /x\\          
   |    sin|-||          
   |       \2/|          
   |1 + ------|*sin(x)   
   |       /x\|          
   |    cos|-||          
   \       \2//          
-------------------------
/       /x\\             
|    sin|-||             
|       \2/|             
|1 - ------|*(1 + sin(x))
|       /x\|             
|    cos|-||             
\       \2//
(sin⁡(x2)cos⁡(x2)+1)sin⁡(x)(−sin⁡(x2)cos⁡(x2)+1)(sin⁡(x)+1)\frac{\left(\frac{\sin{\left(\frac{x}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}} + 1\right) \sin{\left(x \right)}}{\left(- \frac{\sin{\left(\frac{x}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}} + 1\right) \left(\sin{\left(x \right)} + 1\right)} 
  /       /x   pi\\                   
  |    cos|- - --||                   
  |       \2   2 /|    /x\    /x   pi\
2*|1 + -----------|*cos|-|*cos|- - --|
  |          /x\  |    \2/    \2   2 /
  |       cos|-|  |                   
  \          \2/  /                   
--------------------------------------
 /       /x   pi\\                    
 |    cos|- - --||                    
 |       \2   2 /| /       /    pi\\  
 |1 - -----------|*|1 + cos|x - --||  
 |          /x\  | \       \    2 //  
 |       cos|-|  |                    
 \          \2/  /
2(1+cos⁡(x2−π2)cos⁡(x2))cos⁡(x2)cos⁡(x2−π2)(1−cos⁡(x2−π2)cos⁡(x2))(cos⁡(x−π2)+1)\frac{2 \left(1 + \frac{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}}\right) \cos{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 - \frac{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}}\right) \left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right)} 
  /       /x\\    /x\    /x\
2*|1 + tan|-||*cos|-|*sin|-|
  \       \2//    \2/    \2/
----------------------------
 /       /x\\               
 |1 - tan|-||*(1 + sin(x))  
 \       \2//
2(tan⁡(x2)+1)sin⁡(x2)cos⁡(x2)(1−tan⁡(x2))(sin⁡(x)+1)\frac{2 \left(\tan{\left(\frac{x}{2} \right)} + 1\right) \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}}{\left(1 - \tan{\left(\frac{x}{2} \right)}\right) \left(\sin{\left(x \right)} + 1\right)} 
                  /x\           
               sec|-|           
                  \2/           
           1 + ------           
                  /x\           
               csc|-|           
                  \2/           
--------------------------------
             /       /x\\       
             |    sec|-||       
/      1   \ |       \2/|       
|1 + ------|*|1 - ------|*csc(x)
\    csc(x)/ |       /x\|       
             |    csc|-||       
             \       \2//
1+sec⁡(x2)csc⁡(x2)(1−sec⁡(x2)csc⁡(x2))(1+1csc⁡(x))csc⁡(x)\frac{1 + \frac{\sec{\left(\frac{x}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}}}{\left(1 - \frac{\sec{\left(\frac{x}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}}\right) \left(1 + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)}} 
             2/x\ /      1   \    /x\        
        8*cot |-|*|1 + ------|*cot|-|        
              \4/ |       /x\|    \2/        
                  |    cot|-||               
                  \       \2//               
---------------------------------------------
                            /           /x\ \
             2              |      2*cot|-| |
/       2/x\\  /      1   \ |           \2/ |
|1 + cot |-|| *|1 - ------|*|1 + -----------|
\        \4//  |       /x\| |           2/x\|
               |    cot|-|| |    1 + cot |-||
               \       \2// \            \2//
8(1+1cot⁡(x2))cot⁡2(x4)cot⁡(x2)(1+2cot⁡(x2)cot⁡2(x2)+1)(1−1cot⁡(x2))(cot⁡2(x4)+1)2\frac{8 \left(1 + \frac{1}{\cot{\left(\frac{x}{2} \right)}}\right) \cot^{2}{\left(\frac{x}{4} \right)} \cot{\left(\frac{x}{2} \right)}}{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(1 - \frac{1}{\cot{\left(\frac{x}{2} \right)}}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}} 
   /      1        1   \          
   |1 + ------ - ------|*sin(x)   
   \    sin(x)   tan(x)/          
----------------------------------
             /      1        1   \
(1 + sin(x))*|1 + ------ - ------|
             \    tan(x)   sin(x)/
(1−1tan⁡(x)+1sin⁡(x))sin⁡(x)(sin⁡(x)+1)(1+1tan⁡(x)−1sin⁡(x))\frac{\left(1 - \frac{1}{\tan{\left(x \right)}} + \frac{1}{\sin{\left(x \right)}}\right) \sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(1 + \frac{1}{\tan{\left(x \right)}} - \frac{1}{\sin{\left(x \right)}}\right)} 
      2/x\    2/x\ /      1        1   \ 
 4*cos |-|*sin |-|*|1 + ------ - ------| 
       \2/     \2/ \    sin(x)   tan(x)/ 
-----------------------------------------
             /      1        1   \       
(1 + sin(x))*|1 + ------ - ------|*sin(x)
             \    tan(x)   sin(x)/
4(1−1tan⁡(x)+1sin⁡(x))sin⁡2(x2)cos⁡2(x2)(sin⁡(x)+1)(1+1tan⁡(x)−1sin⁡(x))sin⁡(x)\frac{4 \left(1 - \frac{1}{\tan{\left(x \right)}} + \frac{1}{\sin{\left(x \right)}}\right) \sin^{2}{\left(\frac{x}{2} \right)} \cos^{2}{\left(\frac{x}{2} \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(1 + \frac{1}{\tan{\left(x \right)}} - \frac{1}{\sin{\left(x \right)}}\right) \sin{\left(x \right)}} 
             /                    2/x\\                   
             |             1 + cot |-||                   
             |                     \2/|    /x\            
           2*|1 - cot(x) + -----------|*cot|-|            
             |                    /x\ |    \2/            
             |               2*cot|-| |                   
             \                    \2/ /                   
----------------------------------------------------------
              /           /x\ \ /           2/x\         \
              |      2*cot|-| | |    1 + cot |-|         |
/       2/x\\ |           \2/ | |            \2/         |
|1 + cot |-||*|1 + -----------|*|1 - ----------- + cot(x)|
\        \2// |           2/x\| |           /x\          |
              |    1 + cot |-|| |      2*cot|-|          |
              \            \2// \           \2/          /
2(cot⁡2(x2)+12cot⁡(x2)−cot⁡(x)+1)cot⁡(x2)(1+2cot⁡(x2)cot⁡2(x2)+1)(cot⁡2(x2)+1)(−cot⁡2(x2)+12cot⁡(x2)+cot⁡(x)+1)\frac{2 \left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} + 1}{2 \cot{\left(\frac{x}{2} \right)}} - \cot{\left(x \right)} + 1\right) \cot{\left(\frac{x}{2} \right)}}{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)} + 1}{2 \cot{\left(\frac{x}{2} \right)}} + \cot{\left(x \right)} + 1\right)} 
   /      1            \          
   |1 - ------ + csc(x)|*sin(x)   
   \    tan(x)         /          
----------------------------------
             /      1            \
(1 + sin(x))*|1 + ------ - csc(x)|
             \    tan(x)         /
(csc⁡(x)+1−1tan⁡(x))sin⁡(x)(sin⁡(x)+1)(−csc⁡(x)+1+1tan⁡(x))\frac{\left(\csc{\left(x \right)} + 1 - \frac{1}{\tan{\left(x \right)}}\right) \sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(- \csc{\left(x \right)} + 1 + \frac{1}{\tan{\left(x \right)}}\right)} 
                    2                        
       /       2/x\\  /       /x\\    /x\    
     2*|1 - tan |-|| *|1 + tan|-||*tan|-|    
       \        \4//  \       \2//    \2/    
---------------------------------------------
                            /           /x\ \
             2              |      2*tan|-| |
/       2/x\\  /       /x\\ |           \2/ |
|1 + tan |-|| *|1 - tan|-||*|1 + -----------|
\        \4//  \       \2// |           2/x\|
                            |    1 + tan |-||
                            \            \2//
2(1−tan⁡2(x4))2(tan⁡(x2)+1)tan⁡(x2)(1+2tan⁡(x2)tan⁡2(x2)+1)(1−tan⁡(x2))(tan⁡2(x4)+1)2\frac{2 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{2} \left(\tan{\left(\frac{x}{2} \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}}{\left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(1 - \tan{\left(\frac{x}{2} \right)}\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}} 
                       /         2/x\\
                       |    2*sin |-||
     2/x\    2/pi   x\ |          \2/|
4*sin |-|*sin |-- + -|*|1 + ---------|
      \2/     \2    2/ \      sin(x) /
--------------------------------------
 /         2/x\\                      
 |    2*sin |-||                      
 |          \2/|                      
 |1 - ---------|*(1 + sin(x))*sin(x)  
 \      sin(x) /
4(2sin⁡2(x2)sin⁡(x)+1)sin⁡2(x2)sin⁡2(x2+π2)(−2sin⁡2(x2)sin⁡(x)+1)(sin⁡(x)+1)sin⁡(x)\frac{4 \left(\frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} + 1\right) \sin^{2}{\left(\frac{x}{2} \right)} \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}}{\left(- \frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{\sin{\left(x \right)}} + 1\right) \left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}} 
   /       /x   pi\\               
   |    cos|- - --||               
   |       \2   2 /|    /    pi\   
   |1 + -----------|*cos|x - --|   
   |          /x\  |    \    2 /   
   |       cos|-|  |               
   \          \2/  /               
-----------------------------------
/       /x   pi\\                  
|    cos|- - --||                  
|       \2   2 /| /       /    pi\\
|1 - -----------|*|1 + cos|x - --||
|          /x\  | \       \    2 //
|       cos|-|  |                  
\          \2/  /
(1+cos⁡(x2−π2)cos⁡(x2))cos⁡(x−π2)(1−cos⁡(x2−π2)cos⁡(x2))(cos⁡(x−π2)+1)\frac{\left(1 + \frac{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}}\right) \cos{\left(x - \frac{\pi}{2} \right)}}{\left(1 - \frac{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{x}{2} \right)}}\right) \left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right)} 
     2/x\ /      1            \    /x\
2*sin |-|*|1 + ------ - cot(x)|*cot|-|
      \2/ \    sin(x)         /    \2/
--------------------------------------
               /      1            \  
  (1 + sin(x))*|1 - ------ + cot(x)|  
               \    sin(x)         /
2(−cot⁡(x)+1+1sin⁡(x))sin⁡2(x2)cot⁡(x2)(sin⁡(x)+1)(cot⁡(x)+1−1sin⁡(x))\frac{2 \left(- \cot{\left(x \right)} + 1 + \frac{1}{\sin{\left(x \right)}}\right) \sin^{2}{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\cot{\left(x \right)} + 1 - \frac{1}{\sin{\left(x \right)}}\right)} 
                  csc(x)                      
           1 - ----------- + csc(x)           
                  /pi    \                    
               csc|-- - x|                    
                  \2     /                    
----------------------------------------------
/      1   \ /                csc(x)  \       
|1 + ------|*|1 - csc(x) + -----------|*csc(x)
\    csc(x)/ |                /pi    \|       
             |             csc|-- - x||       
             \                \2     //
csc⁡(x)−csc⁡(x)csc⁡(−x+π2)+1(1+1csc⁡(x))(−csc⁡(x)+csc⁡(x)csc⁡(−x+π2)+1)csc⁡(x)\frac{\csc{\left(x \right)} - \frac{\csc{\left(x \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}} + 1}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right) \left(- \csc{\left(x \right)} + \frac{\csc{\left(x \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(x \right)}} 
      4/x\    2/x\ /       /x\\ 
 8*cos |-|*tan |-|*|1 + tan|-|| 
       \4/     \4/ \       \2// 
--------------------------------
/       /x\\                 /x\
|1 - tan|-||*(1 + sin(x))*tan|-|
\       \2//                 \2/
8(tan⁡(x2)+1)cos⁡4(x4)tan⁡2(x4)(1−tan⁡(x2))(sin⁡(x)+1)tan⁡(x2)\frac{8 \left(\tan{\left(\frac{x}{2} \right)} + 1\right) \cos^{4}{\left(\frac{x}{4} \right)} \tan^{2}{\left(\frac{x}{4} \right)}}{\left(1 - \tan{\left(\frac{x}{2} \right)}\right) \left(\sin{\left(x \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}} 
             /      1   \    /x\            
           2*|1 + ------|*cot|-|            
             |       /x\|    \2/            
             |    cot|-||                   
             \       \2//                   
--------------------------------------------
                           /           /x\ \
                           |      2*cot|-| |
/       2/x\\ /      1   \ |           \2/ |
|1 + cot |-||*|1 - ------|*|1 + -----------|
\        \2// |       /x\| |           2/x\|
              |    cot|-|| |    1 + cot |-||
              \       \2// \            \2//
2(1+1cot⁡(x2))cot⁡(x2)(1+2cot⁡(x2)cot⁡2(x2)+1)(1−1cot⁡(x2))(cot⁡2(x2)+1)\frac{2 \left(1 + \frac{1}{\cot{\left(\frac{x}{2} \right)}}\right) \cot{\left(\frac{x}{2} \right)}}{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(1 - \frac{1}{\cot{\left(\frac{x}{2} \right)}}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
                   /          /x\  \                  
                   |       sec|-|  |                  
                   |          \2/  |                  
                 2*|1 + -----------|                  
                   |       /x   pi\|                  
                   |    sec|- - --||                  
                   \       \2   2 //                  
------------------------------------------------------
                  /          /x\  \                   
                  |       sec|-|  |                   
/         1     \ |          \2/  |    /x\    /x   pi\
|1 + -----------|*|1 - -----------|*sec|-|*sec|- - --|
|       /    pi\| |       /x   pi\|    \2/    \2   2 /
|    sec|x - --|| |    sec|- - --||                   
\       \    2 // \       \2   2 //
2(sec⁡(x2)sec⁡(x2−π2)+1)(1+1sec⁡(x−π2))(−sec⁡(x2)sec⁡(x2−π2)+1)sec⁡(x2)sec⁡(x2−π2)\frac{2 \left(\frac{\sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)}{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \left(- \frac{\sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{x}{2} \right)} \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} 
               2                            
  /       2/x\\     4/x\ /       /x\\    /x\
2*|1 - tan |-|| *cos |-|*|1 + tan|-||*tan|-|
  \        \4//      \4/ \       \2//    \2/
--------------------------------------------
         /       /x\\                       
         |1 - tan|-||*(1 + sin(x))          
         \       \2//
2(1−tan⁡2(x4))2(tan⁡(x2)+1)cos⁡4(x4)tan⁡(x2)(1−tan⁡(x2))(sin⁡(x)+1)\frac{2 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{2} \left(\tan{\left(\frac{x}{2} \right)} + 1\right) \cos^{4}{\left(\frac{x}{4} \right)} \tan{\left(\frac{x}{2} \right)}}{\left(1 - \tan{\left(\frac{x}{2} \right)}\right) \left(\sin{\left(x \right)} + 1\right)} 
                          /x\                  
                       sec|-|                  
                          \2/                  
                1 + -----------                
                       /x   pi\                
                    sec|- - --|                
                       \2   2 /                
-----------------------------------------------
                  /          /x\  \            
                  |       sec|-|  |            
/         1     \ |          \2/  |    /    pi\
|1 + -----------|*|1 - -----------|*sec|x - --|
|       /    pi\| |       /x   pi\|    \    2 /
|    sec|x - --|| |    sec|- - --||            
\       \    2 // \       \2   2 //
sec⁡(x2)sec⁡(x2−π2)+1(1+1sec⁡(x−π2))(−sec⁡(x2)sec⁡(x2−π2)+1)sec⁡(x−π2)\frac{\frac{\sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1}{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \left(- \frac{\sec{\left(\frac{x}{2} \right)}}{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(x - \frac{\pi}{2} \right)}} 
     2/x\    4/x\ /       /x\\    /x\
8*cot |-|*sin |-|*|1 + tan|-||*cot|-|
      \4/     \4/ \       \2//    \2/
-------------------------------------
      /       /x\\                   
      |1 - tan|-||*(1 + sin(x))      
      \       \2//
8(tan⁡(x2)+1)sin⁡4(x4)cot⁡2(x4)cot⁡(x2)(1−tan⁡(x2))(sin⁡(x)+1)\frac{8 \left(\tan{\left(\frac{x}{2} \right)} + 1\right) \sin^{4}{\left(\frac{x}{4} \right)} \cot^{2}{\left(\frac{x}{4} \right)} \cot{\left(\frac{x}{2} \right)}}{\left(1 - \tan{\left(\frac{x}{2} \right)}\right) \left(\sin{\left(x \right)} + 1\right)} 
             /                    2/x\\                   
             |             1 + tan |-||                   
             |      1              \2/|    /x\            
           2*|1 - ------ + -----------|*tan|-|            
             |    tan(x)          /x\ |    \2/            
             |               2*tan|-| |                   
             \                    \2/ /                   
----------------------------------------------------------
              /           /x\ \ /                    2/x\\
              |      2*tan|-| | |             1 + tan |-||
/       2/x\\ |           \2/ | |      1              \2/|
|1 + tan |-||*|1 + -----------|*|1 + ------ - -----------|
\        \2// |           2/x\| |    tan(x)          /x\ |
              |    1 + tan |-|| |               2*tan|-| |
              \            \2// \                    \2/ /
2(tan⁡2(x2)+12tan⁡(x2)+1−1tan⁡(x))tan⁡(x2)(1+2tan⁡(x2)tan⁡2(x2)+1)(tan⁡2(x2)+1)(−tan⁡2(x2)+12tan⁡(x2)+1+1tan⁡(x))\frac{2 \left(\frac{\tan^{2}{\left(\frac{x}{2} \right)} + 1}{2 \tan{\left(\frac{x}{2} \right)}} + 1 - \frac{1}{\tan{\left(x \right)}}\right) \tan{\left(\frac{x}{2} \right)}}{\left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(- \frac{\tan^{2}{\left(\frac{x}{2} \right)} + 1}{2 \tan{\left(\frac{x}{2} \right)}} + 1 + \frac{1}{\tan{\left(x \right)}}\right)} 
                  /pi   x\           
               csc|-- - -|           
                  \2    2/           
           1 + -----------           
                     /x\             
                  csc|-|             
                     \2/             
-------------------------------------
             /       /pi   x\\       
             |    csc|-- - -||       
/      1   \ |       \2    2/|       
|1 + ------|*|1 - -----------|*csc(x)
\    csc(x)/ |          /x\  |       
             |       csc|-|  |       
             \          \2/  /
1+csc⁡(−x2+π2)csc⁡(x2)(1−csc⁡(−x2+π2)csc⁡(x2))(1+1csc⁡(x))csc⁡(x)\frac{1 + \frac{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}}}{\left(1 - \frac{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}}\right) \left(1 + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)}} 
     2/x\ /       /x\\    /x\
2*cos |-|*|1 + tan|-||*tan|-|
      \2/ \       \2//    \2/
-----------------------------
  /       /x\\               
  |1 - tan|-||*(1 + sin(x))  
  \       \2//
2(tan⁡(x2)+1)cos⁡2(x2)tan⁡(x2)(1−tan⁡(x2))(sin⁡(x)+1)\frac{2 \left(\tan{\left(\frac{x}{2} \right)} + 1\right) \cos^{2}{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} \right)}}{\left(1 - \tan{\left(\frac{x}{2} \right)}\right) \left(\sin{\left(x \right)} + 1\right)} 
     2/x\ /      1        1   \    /x\
2*cos |-|*|1 + ------ - ------|*tan|-|
      \2/ \    sin(x)   tan(x)/    \2/
--------------------------------------
               /      1        1   \  
  (1 + sin(x))*|1 + ------ - ------|  
               \    tan(x)   sin(x)/
2(1−1tan⁡(x)+1sin⁡(x))cos⁡2(x2)tan⁡(x2)(sin⁡(x)+1)(1+1tan⁡(x)−1sin⁡(x))\frac{2 \left(1 - \frac{1}{\tan{\left(x \right)}} + \frac{1}{\sin{\left(x \right)}}\right) \cos^{2}{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(1 + \frac{1}{\tan{\left(x \right)}} - \frac{1}{\sin{\left(x \right)}}\right)} 
     2/x\ /       /x\\    /x\
2*sin |-|*|1 + tan|-||*cot|-|
      \2/ \       \2//    \2/
-----------------------------
  /       /x\\               
  |1 - tan|-||*(1 + sin(x))  
  \       \2//
2(tan⁡(x2)+1)sin⁡2(x2)cot⁡(x2)(1−tan⁡(x2))(sin⁡(x)+1)\frac{2 \left(\tan{\left(\frac{x}{2} \right)} + 1\right) \sin^{2}{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} \right)}}{\left(1 - \tan{\left(\frac{x}{2} \right)}\right) \left(\sin{\left(x \right)} + 1\right)} 
                           2                        
             /        2/x\\  /      1   \           
           2*|-1 + cot |-|| *|1 + ------|           
             \         \4//  |       /x\|           
                             |    cot|-||           
                             \       \2//           
----------------------------------------------------
                            /           /x\ \       
             2              |      2*cot|-| |       
/       2/x\\  /      1   \ |           \2/ |    /x\
|1 + cot |-|| *|1 - ------|*|1 + -----------|*cot|-|
\        \4//  |       /x\| |           2/x\|    \2/
               |    cot|-|| |    1 + cot |-||       
               \       \2// \            \2//
2(1+1cot⁡(x2))(cot⁡2(x4)−1)2(1+2cot⁡(x2)cot⁡2(x2)+1)(1−1cot⁡(x2))(cot⁡2(x4)+1)2cot⁡(x2)\frac{2 \left(1 + \frac{1}{\cot{\left(\frac{x}{2} \right)}}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{2}}{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(1 - \frac{1}{\cot{\left(\frac{x}{2} \right)}}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \cot{\left(\frac{x}{2} \right)}} 
                2                            
  /        2/x\\     4/x\ /       /x\\    /x\
2*|-1 + cot |-|| *sin |-|*|1 + tan|-||*tan|-|
  \         \4//      \4/ \       \2//    \2/
---------------------------------------------
          /       /x\\                       
          |1 - tan|-||*(1 + sin(x))          
          \       \2//
2(tan⁡(x2)+1)(cot⁡2(x4)−1)2sin⁡4(x4)tan⁡(x2)(1−tan⁡(x2))(sin⁡(x)+1)\frac{2 \left(\tan{\left(\frac{x}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{2} \sin^{4}{\left(\frac{x}{4} \right)} \tan{\left(\frac{x}{2} \right)}}{\left(1 - \tan{\left(\frac{x}{2} \right)}\right) \left(\sin{\left(x \right)} + 1\right)} 
   /         1           cos(x)  \    /    pi\   
   |1 + ----------- - -----------|*cos|x - --|   
   |       /    pi\      /    pi\|    \    2 /   
   |    cos|x - --|   cos|x - --||               
   \       \    2 /      \    2 //               
-------------------------------------------------
/       /    pi\\ /         1           cos(x)  \
|1 + cos|x - --||*|1 - ----------- + -----------|
\       \    2 // |       /    pi\      /    pi\|
                  |    cos|x - --|   cos|x - --||
                  \       \    2 /      \    2 //
(−cos⁡(x)cos⁡(x−π2)+1+1cos⁡(x−π2))cos⁡(x−π2)(cos⁡(x−π2)+1)(cos⁡(x)cos⁡(x−π2)+1−1cos⁡(x−π2))\frac{\left(- \frac{\cos{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}} + 1 + \frac{1}{\cos{\left(x - \frac{\pi}{2} \right)}}\right) \cos{\left(x - \frac{\pi}{2} \right)}}{\left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right) \left(\frac{\cos{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}} + 1 - \frac{1}{\cos{\left(x - \frac{\pi}{2} \right)}}\right)} 
   /      1       sin(2*x)\          
   |1 + ------ - ---------|*sin(x)   
   |    sin(x)        2   |          
   \             2*sin (x)/          
-------------------------------------
             /      1       sin(2*x)\
(1 + sin(x))*|1 - ------ + ---------|
             |    sin(x)        2   |
             \             2*sin (x)/
(1+1sin⁡(x)−sin⁡(2x)2sin⁡2(x))sin⁡(x)(sin⁡(x)+1)(1−1sin⁡(x)+sin⁡(2x)2sin⁡2(x))\frac{\left(1 + \frac{1}{\sin{\left(x \right)}} - \frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(x \right)}}\right) \sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(1 - \frac{1}{\sin{\left(x \right)}} + \frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(x \right)}}\right)} 
                 /       /pi   x\\               
                 |    csc|-- - -||               
                 |       \2    2/|               
               2*|1 + -----------|               
                 |          /x\  |               
                 |       csc|-|  |               
                 \          \2/  /               
-------------------------------------------------
             /       /pi   x\\                   
             |    csc|-- - -||                   
/      1   \ |       \2    2/|    /x\    /pi   x\
|1 + ------|*|1 - -----------|*csc|-|*csc|-- - -|
\    csc(x)/ |          /x\  |    \2/    \2    2/
             |       csc|-|  |                   
             \          \2/  /
2(1+csc⁡(−x2+π2)csc⁡(x2))(1−csc⁡(−x2+π2)csc⁡(x2))(1+1csc⁡(x))csc⁡(x2)csc⁡(−x2+π2)\frac{2 \left(1 + \frac{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}}\right)}{\left(1 - \frac{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{x}{2} \right)}}\right) \left(1 + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(\frac{x}{2} \right)} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} 
   /       /x\\          
   |1 + tan|-||*sin(x)   
   \       \2//          
-------------------------
/       /x\\             
|1 - tan|-||*(1 + sin(x))
\       \2//
(tan⁡(x2)+1)sin⁡(x)(1−tan⁡(x2))(sin⁡(x)+1)\frac{\left(\tan{\left(\frac{x}{2} \right)} + 1\right) \sin{\left(x \right)}}{\left(1 - \tan{\left(\frac{x}{2} \right)}\right) \left(\sin{\left(x \right)} + 1\right)} 
                       /    pi\                              
                    sec|x - --|                              
                       \    2 /      /    pi\                
                1 - ----------- + sec|x - --|                
                       sec(x)        \    2 /                
-------------------------------------------------------------
                  /                     /    pi\\            
                  |                  sec|x - --||            
/         1     \ |       /    pi\      \    2 /|    /    pi\
|1 + -----------|*|1 - sec|x - --| + -----------|*sec|x - --|
|       /    pi\| \       \    2 /      sec(x)  /    \    2 /
|    sec|x - --||                                            
\       \    2 //
sec⁡(x−π2)+1−sec⁡(x−π2)sec⁡(x)(1+1sec⁡(x−π2))(−sec⁡(x−π2)+1+sec⁡(x−π2)sec⁡(x))sec⁡(x−π2)\frac{\sec{\left(x - \frac{\pi}{2} \right)} + 1 - \frac{\sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}}}{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \left(- \sec{\left(x - \frac{\pi}{2} \right)} + 1 + \frac{\sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}} 
(1 - sec(x - pi/2)/sec(x) + sec(x - pi/2))/((1 + 1/sec(x - pi/2))*(1 - sec(x - pi/2) + sec(x - pi/2)/sec(x))*sec(x - pi/2))
    © Sr Examen – Калькулятор