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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
  sin(x)     (1 - cos(x))*cos(x)
---------- - -------------------
1 + sin(x)                  2   
                (1 + sin(x))
−(1−cos⁡(x))cos⁡(x)(sin⁡(x)+1)2+sin⁡(x)sin⁡(x)+1- \frac{\left(1 - \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}} + \frac{\sin{\left(x \right)}}{\sin{\left(x \right)} + 1} 
sin(x)/(1 + sin(x)) - (1 - cos(x))*cos(x)/(1 + sin(x))^2
Simplificación general [src] 
      ___    /    pi\
1 - \/ 2 *cos|x + --|
             \    4 /
---------------------
                2    
    (1 + sin(x))
−2cos⁡(x+π4)+1(sin⁡(x)+1)2\frac{- \sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)} + 1}{\left(\sin{\left(x \right)} + 1\right)^{2}} 
(1 - sqrt(2)*cos(x + pi/4))/(1 + sin(x))^2
Respuesta numérica [src] 
sin(x)/(1.0 + sin(x)) - (1.0 - cos(x))*cos(x)/(1.0 + sin(x))^2
sin(x)/(1.0 + sin(x)) - (1.0 - cos(x))*cos(x)/(1.0 + sin(x))^2
Denominador racional [src] 
            2                                          
(1 + sin(x)) *sin(x) - (1 - cos(x))*(1 + sin(x))*cos(x)
-------------------------------------------------------
                                 3                     
                     (1 + sin(x))
−(1−cos⁡(x))(sin⁡(x)+1)cos⁡(x)+(sin⁡(x)+1)2sin⁡(x)(sin⁡(x)+1)3\frac{- \left(1 - \cos{\left(x \right)}\right) \left(\sin{\left(x \right)} + 1\right) \cos{\left(x \right)} + \left(\sin{\left(x \right)} + 1\right)^{2} \sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{3}} 
((1 + sin(x))^2*sin(x) - (1 - cos(x))*(1 + sin(x))*cos(x))/(1 + sin(x))^3
Denominador común [src] 
            2                     
    -1 + cos (x) - cos(x) - sin(x)
1 + ------------------------------
               2                  
        1 + sin (x) + 2*sin(x)
1+−sin⁡(x)+cos⁡2(x)−cos⁡(x)−1sin⁡2(x)+2sin⁡(x)+11 + \frac{- \sin{\left(x \right)} + \cos^{2}{\left(x \right)} - \cos{\left(x \right)} - 1}{\sin^{2}{\left(x \right)} + 2 \sin{\left(x \right)} + 1} 
1 + (-1 + cos(x)^2 - cos(x) - sin(x))/(1 + sin(x)^2 + 2*sin(x))
Combinatoria [src] 
   2         2                     
cos (x) + sin (x) - cos(x) + sin(x)
-----------------------------------
                       2           
           (1 + sin(x))
sin⁡2(x)+sin⁡(x)+cos⁡2(x)−cos⁡(x)(sin⁡(x)+1)2\frac{\sin^{2}{\left(x \right)} + \sin{\left(x \right)} + \cos^{2}{\left(x \right)} - \cos{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}} 
(cos(x)^2 + sin(x)^2 - cos(x) + sin(x))/(1 + sin(x))^2
Unión de expresiones racionales [src] 
(1 + sin(x))*sin(x) - (1 - cos(x))*cos(x)
-----------------------------------------
                          2              
              (1 + sin(x))
−(1−cos⁡(x))cos⁡(x)+(sin⁡(x)+1)sin⁡(x)(sin⁡(x)+1)2\frac{- \left(1 - \cos{\left(x \right)}\right) \cos{\left(x \right)} + \left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}} 
((1 + sin(x))*sin(x) - (1 - cos(x))*cos(x))/(1 + sin(x))^2
Abrimos la expresión [src] 
          2                                                 
       cos (x)             sin(x)             cos(x)        
---------------------- + ---------- - ----------------------
       2                 1 + sin(x)          2              
1 + sin (x) + 2*sin(x)                1 + sin (x) + 2*sin(x)
cos⁡2(x)sin⁡2(x)+2sin⁡(x)+1−cos⁡(x)sin⁡2(x)+2sin⁡(x)+1+sin⁡(x)sin⁡(x)+1\frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)} + 2 \sin{\left(x \right)} + 1} - \frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)} + 2 \sin{\left(x \right)} + 1} + \frac{\sin{\left(x \right)}}{\sin{\left(x \right)} + 1} 
cos(x)^2/(1 + sin(x)^2 + 2*sin(x)) + sin(x)/(1 + sin(x)) - cos(x)/(1 + sin(x)^2 + 2*sin(x))
Potencias [src] 
  / I*x    -I*x\ /     I*x    -I*x\                             
  |e      e    | |    e      e    |                             
  |---- + -----|*|1 - ---- - -----|         /   -I*x    I*x\    
  \ 2       2  / \     2       2  /       I*\- e     + e   /    
- --------------------------------- - --------------------------
                              2         /      /   -I*x    I*x\\
      /      /   -I*x    I*x\\          |    I*\- e     + e   /|
      |    I*\- e     + e   /|        2*|1 - ------------------|
      |1 - ------------------|          \            2         /
      \            2         /
−i(eix−e−ix)2(−i(eix−e−ix)2+1)−(eix2+e−ix2)(−eix2+1−e−ix2)(−i(eix−e−ix)2+1)2- \frac{i \left(e^{i x} - e^{- i x}\right)}{2 \left(- \frac{i \left(e^{i x} - e^{- i x}\right)}{2} + 1\right)} - \frac{\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right) \left(- \frac{e^{i x}}{2} + 1 - \frac{e^{- i x}}{2}\right)}{\left(- \frac{i \left(e^{i x} - e^{- i x}\right)}{2} + 1\right)^{2}} 
-(exp(i*x)/2 + exp(-i*x)/2)*(1 - exp(i*x)/2 - exp(-i*x)/2)/(1 - i*(-exp(-i*x) + exp(i*x))/2)^2 - i*(-exp(-i*x) + exp(i*x))/(2*(1 - i*(-exp(-i*x) + exp(i*x))/2))
Parte trigonométrica [src] 
      ___    /    3*pi\
1 - \/ 2 *sin|x + ----|
             \     4  /
-----------------------
                 2     
     (1 + sin(x))
−2sin⁡(x+3π4)+1(sin⁡(x)+1)2\frac{- \sqrt{2} \sin{\left(x + \frac{3 \pi}{4} \right)} + 1}{\left(\sin{\left(x \right)} + 1\right)^{2}} 
          ___     
        \/ 2      
 1 - -----------  
        /    pi\  
     sec|x + --|  
        \    4 /  
------------------
                 2
/         1     \ 
|1 + -----------| 
|       /    pi\| 
|    sec|x - --|| 
\       \    2 //
1−2sec⁡(x+π4)(1+1sec⁡(x−π2))2\frac{1 - \frac{\sqrt{2}}{\sec{\left(x + \frac{\pi}{4} \right)}}}{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{2}} 
      /       2/x\\       ___    /x   pi\
    2*|1 - tan |-||   2*\/ 2 *tan|- + --|
      \        \2//              \2   8 /
1 - --------------- + -------------------
             2/x\              2/x   pi\ 
      1 + tan |-|       1 + tan |- + --| 
              \2/               \2   8 / 
-----------------------------------------
                             2           
            /           /x\ \            
            |      2*tan|-| |            
            |           \2/ |            
            |1 + -----------|            
            |           2/x\|            
            |    1 + tan |-||            
            \            \2//
−2(1−tan⁡2(x2))tan⁡2(x2)+1+1+22tan⁡(x2+π8)tan⁡2(x2+π8)+1(1+2tan⁡(x2)tan⁡2(x2)+1)2\frac{- \frac{2 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1 + \frac{2 \sqrt{2} \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1}}{\left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}} 
                       ___   
         2           \/ 2    
1 - ----------- + -----------
       /pi    \      /    pi\
    csc|-- - x|   csc|x + --|
       \2     /      \    4 /
-----------------------------
                    2        
        /      1   \         
        |1 + ------|         
        \    csc(x)/
1+2csc⁡(x+π4)−2csc⁡(−x+π2)(1+1csc⁡(x))2\frac{1 + \frac{\sqrt{2}}{\csc{\left(x + \frac{\pi}{4} \right)}} - \frac{2}{\csc{\left(- x + \frac{\pi}{2} \right)}}}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right)^{2}} 
      ___    /    pi\
1 - \/ 2 *cos|x + --|
             \    4 /
---------------------
                   2 
  /       /    pi\\  
  |1 + cos|x - --||  
  \       \    2 //
−2cos⁡(x+π4)+1(cos⁡(x−π2)+1)2\frac{- \sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)} + 1}{\left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right)^{2}} 
     /    pi\                        
  cos|x - --|                        
     \    2 /     (1 - cos(x))*cos(x)
--------------- - -------------------
       /    pi\                     2
1 + cos|x - --|    /       /    pi\\ 
       \    2 /    |1 + cos|x - --|| 
                   \       \    2 //
−(1−cos⁡(x))cos⁡(x)(cos⁡(x−π2)+1)2+cos⁡(x−π2)cos⁡(x−π2)+1- \frac{\left(1 - \cos{\left(x \right)}\right) \cos{\left(x \right)}}{\left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right)^{2}} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x - \frac{\pi}{2} \right)} + 1} 
                                 1        
                           1 - ------     
         1                     sec(x)     
------------------- - --------------------
/      1   \                      2       
|1 + ------|*csc(x)   /      1   \        
\    csc(x)/          |1 + ------| *sec(x)
                      \    csc(x)/
1(1+1csc⁡(x))csc⁡(x)−1−1sec⁡(x)(1+1csc⁡(x))2sec⁡(x)\frac{1}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)}} - \frac{1 - \frac{1}{\sec{\left(x \right)}}}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right)^{2} \sec{\left(x \right)}} 
      ___    /    pi\
1 - \/ 2 *cos|x + --|
             \    4 /
---------------------
                2    
    (1 + sin(x))
−2cos⁡(x+π4)+1(sin⁡(x)+1)2\frac{- \sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)} + 1}{\left(\sin{\left(x \right)} + 1\right)^{2}} 
                  ___   
      2         \/ 2    
1 - ------ + -----------
    sec(x)      /    pi\
             sec|x - --|
                \    4 /
------------------------
                    2   
   /         1     \    
   |1 + -----------|    
   |       /    pi\|    
   |    sec|x - --||    
   \       \    2 //
1+2sec⁡(x−π4)−2sec⁡(x)(1+1sec⁡(x−π2))2\frac{1 + \frac{\sqrt{2}}{\sec{\left(x - \frac{\pi}{4} \right)}} - \frac{2}{\sec{\left(x \right)}}}{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{2}} 
                   ___    2/x   pi\    /x   pi\
1 - 2*cos(x) + 2*\/ 2 *sin |- + --|*cot|- + --|
                           \2   8 /    \2   8 /
-----------------------------------------------
                             2                 
                 (1 + sin(x))
22sin⁡2(x2+π8)cot⁡(x2+π8)−2cos⁡(x)+1(sin⁡(x)+1)2\frac{2 \sqrt{2} \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} \cot{\left(\frac{x}{2} + \frac{\pi}{8} \right)} - 2 \cos{\left(x \right)} + 1}{\left(\sin{\left(x \right)} + 1\right)^{2}} 
                                              1          
                                        1 - ------       
              1                             sec(x)       
----------------------------- - -------------------------
/         1     \    /    pi\                    2       
|1 + -----------|*sec|x - --|   /         1     \        
|       /    pi\|    \    2 /   |1 + -----------| *sec(x)
|    sec|x - --||               |       /    pi\|        
\       \    2 //               |    sec|x - --||        
                                \       \    2 //
−1−1sec⁡(x)(1+1sec⁡(x−π2))2sec⁡(x)+1(1+1sec⁡(x−π2))sec⁡(x−π2)- \frac{1 - \frac{1}{\sec{\left(x \right)}}}{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right)^{2} \sec{\left(x \right)}} + \frac{1}{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}} 
                 ___    /    pi\
1 - 2*cos(x) + \/ 2 *sin|x + --|
                        \    4 /
--------------------------------
                     2          
         (1 + sin(x))
2sin⁡(x+π4)−2cos⁡(x)+1(sin⁡(x)+1)2\frac{\sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)} - 2 \cos{\left(x \right)} + 1}{\left(\sin{\left(x \right)} + 1\right)^{2}} 
      ___ /        2/x   pi\\
    \/ 2 *|-1 + cot |- + --||
          \         \2   8 //
1 - -------------------------
                2/x   pi\    
         1 + cot |- + --|    
                 \2   8 /    
-----------------------------
                       2     
      /           /x\ \      
      |      2*cot|-| |      
      |           \2/ |      
      |1 + -----------|      
      |           2/x\|      
      |    1 + cot |-||      
      \            \2//
−2(cot⁡2(x2+π8)−1)cot⁡2(x2+π8)+1+1(1+2cot⁡(x2)cot⁡2(x2)+1)2\frac{- \frac{\sqrt{2} \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} - 1\right)}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1} + 1}{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}} 
      ___ /       2/x   pi\\
    \/ 2 *|1 - tan |- + --||
          \        \2   8 //
1 - ------------------------
               2/x   pi\    
        1 + tan |- + --|    
                \2   8 /    
----------------------------
                      2     
     /           /x\ \      
     |      2*tan|-| |      
     |           \2/ |      
     |1 + -----------|      
     |           2/x\|      
     |    1 + tan |-||      
     \            \2//
−2(1−tan⁡2(x2+π8))tan⁡2(x2+π8)+1+1(1+2tan⁡(x2)tan⁡2(x2)+1)2\frac{- \frac{\sqrt{2} \left(1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1} + 1}{\left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}} 
      ___ /       2/x   pi\\
    \/ 2 *|1 - tan |- + --||
          \        \2   8 //
1 - ------------------------
               2/x   pi\    
        1 + tan |- + --|    
                \2   8 /    
----------------------------
                   2        
       (1 + sin(x))
−2(1−tan⁡2(x2+π8))tan⁡2(x2+π8)+1+1(sin⁡(x)+1)2\frac{- \frac{\sqrt{2} \left(1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1} + 1}{\left(\sin{\left(x \right)} + 1\right)^{2}} 
                                    1          
                           1 - -----------     
                                  /pi    \     
                               csc|-- - x|     
         1                        \2     /     
------------------- - -------------------------
/      1   \                      2            
|1 + ------|*csc(x)   /      1   \     /pi    \
\    csc(x)/          |1 + ------| *csc|-- - x|
                      \    csc(x)/     \2     /
1(1+1csc⁡(x))csc⁡(x)−1−1csc⁡(−x+π2)(1+1csc⁡(x))2csc⁡(−x+π2)\frac{1}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)}} - \frac{1 - \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right)^{2} \csc{\left(- x + \frac{\pi}{2} \right)}} 
         /    pi\     ___    /    pi\
1 - 2*sin|x + --| + \/ 2 *sin|x + --|
         \    2 /            \    4 /
-------------------------------------
                        2            
            (1 + sin(x))
2sin⁡(x+π4)−2sin⁡(x+π2)+1(sin⁡(x)+1)2\frac{\sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)} - 2 \sin{\left(x + \frac{\pi}{2} \right)} + 1}{\left(\sin{\left(x \right)} + 1\right)^{2}} 
         ___    
       \/ 2     
1 - ------------
       /     pi\
    csc|-x + --|
       \     4 /
----------------
             2  
 /      1   \   
 |1 + ------|   
 \    csc(x)/
1−2csc⁡(−x+π4)(1+1csc⁡(x))2\frac{1 - \frac{\sqrt{2}}{\csc{\left(- x + \frac{\pi}{4} \right)}}}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right)^{2}} 
                   ___    /x   pi\
               2*\/ 2 *tan|- + --|
                          \2   8 /
1 - 2*cos(x) + -------------------
                        2/x   pi\ 
                 1 + tan |- + --| 
                         \2   8 / 
----------------------------------
                      2           
          (1 + sin(x))
−2cos⁡(x)+1+22tan⁡(x2+π8)tan⁡2(x2+π8)+1(sin⁡(x)+1)2\frac{- 2 \cos{\left(x \right)} + 1 + \frac{2 \sqrt{2} \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1}}{\left(\sin{\left(x \right)} + 1\right)^{2}} 
      ___    2/x   pi\ /        2/x   pi\\
1 - \/ 2 *sin |- + --|*|-1 + cot |- + --||
              \2   8 / \         \2   8 //
------------------------------------------
                          2               
              (1 + sin(x))
−2(cot⁡2(x2+π8)−1)sin⁡2(x2+π8)+1(sin⁡(x)+1)2\frac{- \sqrt{2} \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} - 1\right) \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1}{\left(\sin{\left(x \right)} + 1\right)^{2}} 
      /        2/x\\       ___    /x   pi\
    2*|-1 + cot |-||   2*\/ 2 *cot|- + --|
      \         \2//              \2   8 /
1 - ---------------- + -------------------
             2/x\               2/x   pi\ 
      1 + cot |-|        1 + cot |- + --| 
              \2/                \2   8 / 
------------------------------------------
                             2            
            /           /x\ \             
            |      2*cot|-| |             
            |           \2/ |             
            |1 + -----------|             
            |           2/x\|             
            |    1 + cot |-||             
            \            \2//
−2(cot⁡2(x2)−1)cot⁡2(x2)+1+1+22cot⁡(x2+π8)cot⁡2(x2+π8)+1(1+2cot⁡(x2)cot⁡2(x2)+1)2\frac{- \frac{2 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1 + \frac{2 \sqrt{2} \cot{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1}}{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2}} 
                                                /           2/x\\ 
                                                |    1 - tan |-|| 
                                  /       2/x\\ |            \2/| 
                                  |1 - tan |-||*|1 - -----------| 
                 /x\              \        \2// |           2/x\| 
            2*tan|-|                            |    1 + tan |-|| 
                 \2/                            \            \2// 
------------------------------- - --------------------------------
              /           /x\ \                                  2
              |      2*tan|-| |                 /           /x\ \ 
/       2/x\\ |           \2/ |                 |      2*tan|-| | 
|1 + tan |-||*|1 + -----------|   /       2/x\\ |           \2/ | 
\        \2// |           2/x\|   |1 + tan |-||*|1 + -----------| 
              |    1 + tan |-||   \        \2// |           2/x\| 
              \            \2//                 |    1 + tan |-|| 
                                                \            \2//
2tan⁡(x2)(1+2tan⁡(x2)tan⁡2(x2)+1)(tan⁡2(x2)+1)−(1−tan⁡2(x2))(−1−tan⁡2(x2)tan⁡2(x2)+1+1)(1+2tan⁡(x2)tan⁡2(x2)+1)2(tan⁡2(x2)+1)\frac{2 \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)} - \frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(- \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)}{\left(1 + \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
                                  /            2/x\\               
                                  |    -1 + cot |-||               
                                  |             \2/| /        2/x\\
                                  |1 - ------------|*|-1 + cot |-||
                 /x\              |           2/x\ | \         \2//
            2*cot|-|              |    1 + cot |-| |               
                 \2/              \            \2/ /               
------------------------------- - ---------------------------------
              /           /x\ \                                   2
              |      2*cot|-| |                  /           /x\ \ 
/       2/x\\ |           \2/ |                  |      2*cot|-| | 
|1 + cot |-||*|1 + -----------|    /       2/x\\ |           \2/ | 
\        \2// |           2/x\|    |1 + cot |-||*|1 + -----------| 
              |    1 + cot |-||    \        \2// |           2/x\| 
              \            \2//                  |    1 + cot |-|| 
                                                 \            \2//
2cot⁡(x2)(1+2cot⁡(x2)cot⁡2(x2)+1)(cot⁡2(x2)+1)−(−cot⁡2(x2)−1cot⁡2(x2)+1+1)(cot⁡2(x2)−1)(1+2cot⁡(x2)cot⁡2(x2)+1)2(cot⁡2(x2)+1)\frac{2 \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)} - \frac{\left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{\left(1 + \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} 
             /       /    pi\\    /    pi\
             |1 - sin|x + --||*sin|x + --|
  sin(x)     \       \    2 //    \    2 /
---------- - -----------------------------
1 + sin(x)                       2        
                     (1 + sin(x))
−(1−sin⁡(x+π2))sin⁡(x+π2)(sin⁡(x)+1)2+sin⁡(x)sin⁡(x)+1- \frac{\left(1 - \sin{\left(x + \frac{\pi}{2} \right)}\right) \sin{\left(x + \frac{\pi}{2} \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}} + \frac{\sin{\left(x \right)}}{\sin{\left(x \right)} + 1} 
                 ___    /    pi\
1 - 2*cos(x) + \/ 2 *cos|x - --|
                        \    4 /
--------------------------------
                        2       
       /       /    pi\\        
       |1 + cos|x - --||        
       \       \    2 //
−2cos⁡(x)+2cos⁡(x−π4)+1(cos⁡(x−π2)+1)2\frac{- 2 \cos{\left(x \right)} + \sqrt{2} \cos{\left(x - \frac{\pi}{4} \right)} + 1}{\left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right)^{2}} 
(1 - 2*cos(x) + sqrt(2)*cos(x - pi/4))/(1 + cos(x - pi/2))^2
    © Sr Examen – Калькулятор