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-((-1+x^2/(1+x^2))/sqrt(1+x^2)+(1+x/sqrt(1+x^2))^2/(x+sqrt(1+x^2)))/(x+sqrt(1+x^2))
((z-1)*((z*f*r*f)/(1+(z-1)*c*f)))/(z*r)
y+y/x-y+1/(x^2)-x
y/(y+3)^2-y/(y^2)-9
Factorizar el polinomio:
z^2+2*z+26
z^2-169/196
y+y^2/2
y*y/x-x^2+x*y/x-y^2
Descomponer al cuadrado:
y^4-y^2-13
y^4-9*y^2+5
-y^4-9*y^2-4
-y^4+y^2+1
Expresiones idénticas
sinx/(((uno +tgx/ dos)/(uno -tgx/ dos))*(uno +sinx))
seno de x dividir por (((1 más tgx dividir por 2) dividir por (1 menos tgx dividir por 2)) multiplicar por (1 más seno de x))
seno de x dividir por (((uno más tgx dividir por dos) dividir por (uno menos tgx dividir por dos)) multiplicar por (uno más seno de x))
sinx/(((1+tgx/2)/(1-tgx/2))(1+sinx))
sinx/1+tgx/2/1-tgx/21+sinx
sinx dividir por (((1+tgx dividir por 2) dividir por (1-tgx dividir por 2))*(1+sinx))
Expresiones semejantes
sinx/(((1+tgx/2)/(1-tgx/2))*(1-sinx))
sinx/(((1+tg(x/2))/(1-tg(x/2)))*(1+sinx))
sinx/(((1+tgx/2)/(1+tgx/2))*(1+sinx))
sinx/(((1-tgx/2)/(1-tgx/2))*(1+sinx))
Expresiones con funciones
sinx
sinx/(tan((pi/4)-(x/2))*(1+sinx))
sinx/(((1+tg(x/2))/(1-tg(x/2)))*(1+sinx))
sinx/(1-cosx)^2
tgx
tgx/(1-(tgx)^2)+ctgx/(1+(ctgx)^2)
tgx
tgx/(1-(tgx)^2)+ctgx/(1+(ctgx)^2)
sinx
sinx/(tan((pi/4)-(x/2))*(1+sinx))
sinx/(((1+tg(x/2))/(1-tg(x/2)))*(1+sinx))
sinx/(1-cosx)^2

Simplificación de expresiones/ Descomposición en fracciones simples/ sinx/(((1+tgx/2)/(1-tgx/2))*(1+sinx))

¿Cómo vas a descomponer esta sinx/(((1+tgx/2)/(1-tgx/2))*(1+sinx)) expresión en fracciones?

Solución

Ha introducido [src]

         sin(x)        
-----------------------
    tan(x)             
1 + ------             
      2                
----------*(1 + sin(x))
    tan(x)             
1 - ------             
      2

\frac{\sin{\left(x \right)}}{\frac{\frac{\tan{\left(x \right)}}{2} + 1}{- \frac{\tan{\left(x \right)}}{2} + 1} \left(\sin{\left(x \right)} + 1\right)}

sin(x)/((((1 + tan(x)/2)/(1 - tan(x)/2))*(1 + sin(x))))

Simplificación general [src]

  -(-2 + tan(x))*sin(x)  
-------------------------
(1 + sin(x))*(2 + tan(x))

- \frac{\left(\tan{\left(x \right)} - 2\right) \sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\tan{\left(x \right)} + 2\right)}

-(-2 + tan(x))*sin(x)/((1 + sin(x))*(2 + tan(x)))

Respuesta numérica [src]

(1.0 - 0.5*tan(x))*sin(x)/((1.0 + 0.5*tan(x))*(1.0 + sin(x)))

(1.0 - 0.5*tan(x))*sin(x)/((1.0 + 0.5*tan(x))*(1.0 + sin(x)))

Potencias [src]

      /      /   I*x    -I*x\\                     
      |    I*\- e    + e    /| /   -I*x    I*x\    
   -I*|1 - ------------------|*\- e     + e   /    
      |       / I*x    -I*x\ |                     
      \     2*\e    + e    / /                     
---------------------------------------------------
  /      /   -I*x    I*x\\ /      /   I*x    -I*x\\
  |    I*\- e     + e   /| |    I*\- e    + e    /|
2*|1 - ------------------|*|1 + ------------------|
  \            2         / |       / I*x    -I*x\ |
                           \     2*\e    + e    / /

- \frac{i \left(- \frac{i \left(- e^{i x} + e^{- i x}\right)}{2 \left(e^{i x} + e^{- i x}\right)} + 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^{i x} + e^{- i x}\right)}{2 \left(e^{i x} + e^{- i x}\right)} + 1\right)}

   /    tan(x)\          
   |1 - ------|*sin(x)   
   \      2   /          
-------------------------
/    tan(x)\             
|1 + ------|*(1 + sin(x))
\      2   /

\frac{\left(1 - \frac{\tan{\left(x \right)}}{2}\right) \sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\frac{\tan{\left(x \right)}}{2} + 1\right)}

(1 - tan(x)/2)*sin(x)/((1 + tan(x)/2)*(1 + sin(x)))

Combinatoria [src]

  -(-2 + tan(x))*sin(x)  
-------------------------
(1 + sin(x))*(2 + tan(x))

- \frac{\left(\tan{\left(x \right)} - 2\right) \sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\tan{\left(x \right)} + 2\right)}

-(-2 + tan(x))*sin(x)/((1 + sin(x))*(2 + tan(x)))

Denominador común [src]

             2 + 4*sin(x) + tan(x)        
-1 + -------------------------------------
     2 + 2*sin(x) + sin(x)*tan(x) + tan(x)

\frac{4 \sin{\left(x \right)} + \tan{\left(x \right)} + 2}{\sin{\left(x \right)} \tan{\left(x \right)} + 2 \sin{\left(x \right)} + \tan{\left(x \right)} + 2} - 1

-1 + (2 + 4*sin(x) + tan(x))/(2 + 2*sin(x) + sin(x)*tan(x) + tan(x))

Denominador racional [src]

   (4 - 2*tan(x))*sin(x)   
---------------------------
(1 + sin(x))*(4 + 2*tan(x))

\frac{\left(4 - 2 \tan{\left(x \right)}\right) \sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(2 \tan{\left(x \right)} + 4\right)}

(4 - 2*tan(x))*sin(x)/((1 + sin(x))*(4 + 2*tan(x)))

Unión de expresiones racionales [src]

   (2 - tan(x))*sin(x)   
-------------------------
(1 + sin(x))*(2 + tan(x))

\frac{\left(2 - \tan{\left(x \right)}\right) \sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\tan{\left(x \right)} + 2\right)}

(2 - tan(x))*sin(x)/((1 + sin(x))*(2 + tan(x)))

Compilar la expresión [src]

   /    tan(x)\          
   |1 - ------|*sin(x)   
   \      2   /          
-------------------------
/    tan(x)\             
|1 + ------|*(1 + sin(x))
\      2   /

\frac{\left(1 - \frac{\tan{\left(x \right)}}{2}\right) \sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\frac{\tan{\left(x \right)}}{2} + 1\right)}

(1 - tan(x)/2)*sin(x)/((1 + tan(x)/2)*(1 + sin(x)))

Abrimos la expresión [src]

                          sin(x)                         
---------------------------------------------------------
    1          sin(x)         tan(x)       sin(x)*tan(x) 
---------- + ---------- + -------------- + --------------
    tan(x)       tan(x)     /    tan(x)\     /    tan(x)\
1 - ------   1 - ------   2*|1 - ------|   2*|1 - ------|
      2            2        \      2   /     \      2   /

\frac{\sin{\left(x \right)}}{\frac{\sin{\left(x \right)} \tan{\left(x \right)}}{2 \left(1 - \frac{\tan{\left(x \right)}}{2}\right)} + \frac{\sin{\left(x \right)}}{1 - \frac{\tan{\left(x \right)}}{2}} + \frac{\tan{\left(x \right)}}{2 \left(1 - \frac{\tan{\left(x \right)}}{2}\right)} + \frac{1}{1 - \frac{\tan{\left(x \right)}}{2}}}

   /    tan(x)\          
   |1 - ------|*sin(x)   
   \      2   /          
-------------------------
/    tan(x)\             
|1 + ------|*(1 + sin(x))
\      2   /

\frac{\left(- \frac{\tan{\left(x \right)}}{2} + 1\right) \sin{\left(x \right)}}{\left(\frac{\tan{\left(x \right)}}{2} + 1\right) \left(\sin{\left(x \right)} + 1\right)}

(1 - tan(x)/2)*sin(x)/((1 + tan(x)/2)*(1 + sin(x)))

Parte trigonométrica [src]

             /    tan(x)\    /x\            
           4*|1 - ------|*tan|-|            
             \      2   /    \2/            
--------------------------------------------
              /           /x\ \             
              |      2*tan|-| |             
/       2/x\\ |           \2/ |             
|1 + tan |-||*|1 + -----------|*(2 + tan(x))
\        \2// |           2/x\|             
              |    1 + tan |-||             
              \            \2//

\frac{4 \left(1 - \frac{\tan{\left(x \right)}}{2}\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(\tan{\left(x \right)} + 2\right)}

   /     sin(x) \          
   |1 - --------|*sin(x)   
   \    2*cos(x)/          
---------------------------
/     sin(x) \             
|1 + --------|*(1 + sin(x))
\    2*cos(x)/

\frac{\left(- \frac{\sin{\left(x \right)}}{2 \cos{\left(x \right)}} + 1\right) \sin{\left(x \right)}}{\left(\frac{\sin{\left(x \right)}}{2 \cos{\left(x \right)}} + 1\right) \left(\sin{\left(x \right)} + 1\right)}

    /    tan(x)\         
  2*|1 - ------|*sin(x)  
    \      2   /         
-------------------------
(1 + sin(x))*(2 + tan(x))

\frac{2 \left(1 - \frac{\tan{\left(x \right)}}{2}\right) \sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\tan{\left(x \right)} + 2\right)}

      2/x\    2/x\ /    tan(x)\ 
 8*cos |-|*sin |-|*|1 - ------| 
       \2/     \2/ \      2   / 
--------------------------------
(1 + sin(x))*(2 + tan(x))*sin(x)

\frac{8 \left(1 - \frac{\tan{\left(x \right)}}{2}\right) \sin^{2}{\left(\frac{x}{2} \right)} \cos^{2}{\left(\frac{x}{2} \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\tan{\left(x \right)} + 2\right) \sin{\left(x \right)}}

                    2                        
       /       2/x\\  /    tan(x)\    /x\    
     4*|1 - tan |-|| *|1 - ------|*tan|-|    
       \        \4//  \      2   /    \2/    
---------------------------------------------
               /           /x\ \             
             2 |      2*tan|-| |             
/       2/x\\  |           \2/ |             
|1 + tan |-|| *|1 + -----------|*(2 + tan(x))
\        \4//  |           2/x\|             
               |    1 + tan |-||             
               \            \2//

\frac{4 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{2} \left(1 - \frac{\tan{\left(x \right)}}{2}\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}{4} \right)} + 1\right)^{2} \left(\tan{\left(x \right)} + 2\right)}

      4/x\    2/x\ /    tan(x)\ 
16*cos |-|*tan |-|*|1 - ------| 
       \4/     \4/ \      2   / 
--------------------------------
                             /x\
(1 + sin(x))*(2 + tan(x))*tan|-|
                             \2/

\frac{16 \left(1 - \frac{\tan{\left(x \right)}}{2}\right) \cos^{4}{\left(\frac{x}{4} \right)} \tan^{2}{\left(\frac{x}{4} \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\tan{\left(x \right)} + 2\right) \tan{\left(\frac{x}{2} \right)}}

             /    tan(x)\    /x\            
           2*|1 - ------|*tan|-|            
             \      2   /    \2/            
--------------------------------------------
                           /           /x\ \
                           |      2*tan|-| |
/       2/x\\ /    tan(x)\ |           \2/ |
|1 + tan |-||*|1 + ------|*|1 + -----------|
\        \2// \      2   / |           2/x\|
                           |    1 + tan |-||
                           \            \2//

\frac{2 \left(1 - \frac{\tan{\left(x \right)}}{2}\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{\left(x \right)}}{2} + 1\right)}

                       /       2    \
     2/x\    2/pi   x\ |    sin (x) |
8*sin |-|*sin |-- + -|*|1 - --------|
      \2/     \2    2/ \    sin(2*x)/
-------------------------------------
              /         2   \        
              |    2*sin (x)|        
 (1 + sin(x))*|2 + ---------|*sin(x) 
              \     sin(2*x)/

\frac{8 \left(- \frac{\sin^{2}{\left(x \right)}}{\sin{\left(2 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(x \right)}}{\sin{\left(2 x \right)}} + 2\right) \left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}

             /       1    \    /x\            
           2*|1 - --------|*cot|-|            
             \    2*cot(x)/    \2/            
----------------------------------------------
                             /           /x\ \
                             |      2*cot|-| |
/       2/x\\ /       1    \ |           \2/ |
|1 + cot |-||*|1 + --------|*|1 + -----------|
\        \2// \    2*cot(x)/ |           2/x\|
                             |    1 + cot |-||
                             \            \2//

\frac{2 \left(1 - \frac{1}{2 \cot{\left(x \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}{2 \cot{\left(x \right)}}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}

   /       2    \          
   |    sin (x) |          
   |1 - --------|*sin(x)   
   \    sin(2*x)/          
---------------------------
/       2    \             
|    sin (x) |             
|1 + --------|*(1 + sin(x))
\    sin(2*x)/

\frac{\left(- \frac{\sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 1\right) \sin{\left(x \right)}}{\left(\frac{\sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 1\right) \left(\sin{\left(x \right)} + 1\right)}

                  /pi    \           
               csc|-- - x|           
                  \2     /           
           1 - -----------           
                 2*csc(x)            
-------------------------------------
             /       /pi    \\       
             |    csc|-- - x||       
/      1   \ |       \2     /|       
|1 + ------|*|1 + -----------|*csc(x)
\    csc(x)/ \      2*csc(x) /

\frac{1 - \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{2 \csc{\left(x \right)}}}{\left(1 + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{2 \csc{\left(x \right)}}\right) \left(1 + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)}}

                sec(x)            
           1 - --------           
               2*csc(x)           
----------------------------------
/      1   \ /     sec(x) \       
|1 + ------|*|1 + --------|*csc(x)
\    csc(x)/ \    2*csc(x)/

\frac{1 - \frac{\sec{\left(x \right)}}{2 \csc{\left(x \right)}}}{\left(1 + \frac{\sec{\left(x \right)}}{2 \csc{\left(x \right)}}\right) \left(1 + \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)}}

               2                            
  /       2/x\\     4/x\ /    tan(x)\    /x\
4*|1 - tan |-|| *cos |-|*|1 - ------|*tan|-|
  \        \4//      \4/ \      2   /    \2/
--------------------------------------------
         (1 + sin(x))*(2 + tan(x))

\frac{4 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{2} \left(1 - \frac{\tan{\left(x \right)}}{2}\right) \cos^{4}{\left(\frac{x}{4} \right)} \tan{\left(\frac{x}{2} \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\tan{\left(x \right)} + 2\right)}

               /        sec(x)   \             
             2*|1 - -------------|             
               |         /    pi\|             
               |    2*sec|x - --||             
               \         \    2 //             
-----------------------------------------------
/         1     \ /       sec(x)  \    /    pi\
|1 + -----------|*|2 + -----------|*sec|x - --|
|       /    pi\| |       /    pi\|    \    2 /
|    sec|x - --|| |    sec|x - --||            
\       \    2 // \       \    2 //

\frac{2 \left(- \frac{\sec{\left(x \right)}}{2 \sec{\left(x - \frac{\pi}{2} \right)}} + 1\right)}{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + 2\right) \sec{\left(x - \frac{\pi}{2} \right)}}

                        sec(x)                   
                1 - -------------                
                         /    pi\                
                    2*sec|x - --|                
                         \    2 /                
-------------------------------------------------
/         1     \ /        sec(x)   \    /    pi\
|1 + -----------|*|1 + -------------|*sec|x - --|
|       /    pi\| |         /    pi\|    \    2 /
|    sec|x - --|| |    2*sec|x - --||            
\       \    2 // \         \    2 //

\frac{- \frac{\sec{\left(x \right)}}{2 \sec{\left(x - \frac{\pi}{2} \right)}} + 1}{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \left(\frac{\sec{\left(x \right)}}{2 \sec{\left(x - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(x - \frac{\pi}{2} \right)}}

                    2/x\ /    tan(x)\               
              16*tan |-|*|1 - ------|               
                     \4/ \      2   /               
----------------------------------------------------
               /           /x\ \                    
             2 |      2*tan|-| |                    
/       2/x\\  |           \2/ |                 /x\
|1 + tan |-|| *|1 + -----------|*(2 + tan(x))*tan|-|
\        \4//  |           2/x\|                 \2/
               |    1 + tan |-||                    
               \            \2//

\frac{16 \left(1 - \frac{\tan{\left(x \right)}}{2}\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(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \left(\tan{\left(x \right)} + 2\right) \tan{\left(\frac{x}{2} \right)}}

             2/x\ /       1    \    /x\      
       16*cot |-|*|1 - --------|*cot|-|      
              \4/ \    2*cot(x)/    \2/      
---------------------------------------------
               /           /x\ \             
             2 |      2*cot|-| |             
/       2/x\\  |           \2/ | /      1   \
|1 + cot |-|| *|1 + -----------|*|2 + ------|
\        \4//  |           2/x\| \    cot(x)/
               |    1 + cot |-||             
               \            \2//

\frac{16 \left(1 - \frac{1}{2 \cot{\left(x \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(2 + \frac{1}{\cot{\left(x \right)}}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}

                          2                         
            /        2/x\\  /       1    \          
          4*|-1 + cot |-|| *|1 - --------|          
            \         \4//  \    2*cot(x)/          
----------------------------------------------------
               /           /x\ \                    
             2 |      2*cot|-| |                    
/       2/x\\  |           \2/ | /      1   \    /x\
|1 + cot |-|| *|1 + -----------|*|2 + ------|*cot|-|
\        \4//  |           2/x\| \    cot(x)/    \2/
               |    1 + cot |-||                    
               \            \2//

\frac{4 \left(1 - \frac{1}{2 \cot{\left(x \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(2 + \frac{1}{\cot{\left(x \right)}}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2} \cot{\left(\frac{x}{2} \right)}}

                2                            
  /        2/x\\     4/x\ /    tan(x)\    /x\
4*|-1 + cot |-|| *sin |-|*|1 - ------|*tan|-|
  \         \4//      \4/ \      2   /    \2/
---------------------------------------------
          (1 + sin(x))*(2 + tan(x))

\frac{4 \left(1 - \frac{\tan{\left(x \right)}}{2}\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(\sin{\left(x \right)} + 1\right) \left(\tan{\left(x \right)} + 2\right)}

                  /        sec(x)   \                 
                4*|1 - -------------|                 
                  |         /    pi\|                 
                  |    2*sec|x - --||                 
                  \         \    2 //                 
------------------------------------------------------
/         1     \ /       sec(x)  \    /x\    /x   pi\
|1 + -----------|*|2 + -----------|*sec|-|*sec|- - --|
|       /    pi\| |       /    pi\|    \2/    \2   2 /
|    sec|x - --|| |    sec|x - --||                   
\       \    2 // \       \    2 //

\frac{4 \left(- \frac{\sec{\left(x \right)}}{2 \sec{\left(x - \frac{\pi}{2} \right)}} + 1\right)}{\left(1 + \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}}\right) \left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + 2\right) \sec{\left(\frac{x}{2} \right)} \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}

   /       /    pi\\               
   |    cos|x - --||               
   |       \    2 /|    /    pi\   
   |1 - -----------|*cos|x - --|   
   \      2*cos(x) /    \    2 /   
-----------------------------------
/       /    pi\\                  
|    cos|x - --||                  
|       \    2 /| /       /    pi\\
|1 + -----------|*|1 + cos|x - --||
\      2*cos(x) / \       \    2 //

\frac{\left(1 - \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{2 \cos{\left(x \right)}}\right) \cos{\left(x - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{2 \cos{\left(x \right)}}\right) \left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right)}

            /       1    \    /x\           
          4*|1 - --------|*cot|-|           
            \    2*cot(x)/    \2/           
--------------------------------------------
              /           /x\ \             
              |      2*cot|-| |             
/       2/x\\ |           \2/ | /      1   \
|1 + cot |-||*|1 + -----------|*|2 + ------|
\        \2// |           2/x\| \    cot(x)/
              |    1 + cot |-||             
              \            \2//

\frac{4 \left(1 - \frac{1}{2 \cot{\left(x \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(2 + \frac{1}{\cot{\left(x \right)}}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}

                 /       /pi    \\               
                 |    csc|-- - x||               
                 |       \2     /|               
               4*|1 - -----------|               
                 \      2*csc(x) /               
-------------------------------------------------
             /       /pi    \\                   
             |    csc|-- - x||                   
/      1   \ |       \2     /|    /x\    /pi   x\
|1 + ------|*|2 + -----------|*csc|-|*csc|-- - -|
\    csc(x)/ \       csc(x)  /    \2/    \2    2/

\frac{4 \left(1 - \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{2 \csc{\left(x \right)}}\right)}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right) \left(2 + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}\right) \csc{\left(\frac{x}{2} \right)} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}

  -(-2 + tan(x))*sin(x)  
-------------------------
(1 + sin(x))*(2 + tan(x))

- \frac{\left(\tan{\left(x \right)} - 2\right) \sin{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\tan{\left(x \right)} + 2\right)}

    /       /    pi\\              
    |    cos|x - --||              
    |       \    2 /|    /    pi\  
  2*|1 - -----------|*cos|x - --|  
    \      2*cos(x) /    \    2 /  
-----------------------------------
                  /       /    pi\\
                  |    cos|x - --||
/       /    pi\\ |       \    2 /|
|1 + cos|x - --||*|2 + -----------|
\       \    2 // \       cos(x)  /

\frac{2 \left(1 - \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{2 \cos{\left(x \right)}}\right) \cos{\left(x - \frac{\pi}{2} \right)}}{\left(2 + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}\right) \left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right)}

     2/x\ /    tan(x)\    /x\
4*cos |-|*|1 - ------|*tan|-|
      \2/ \      2   /    \2/
-----------------------------
  (1 + sin(x))*(2 + tan(x))

\frac{4 \left(1 - \frac{\tan{\left(x \right)}}{2}\right) \cos^{2}{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\tan{\left(x \right)} + 2\right)}

           /       /pi    \\         
           |    csc|-- - x||         
           |       \2     /|         
         2*|1 - -----------|         
           \      2*csc(x) /         
-------------------------------------
             /       /pi    \\       
             |    csc|-- - x||       
/      1   \ |       \2     /|       
|1 + ------|*|2 + -----------|*csc(x)
\    csc(x)/ \       csc(x)  /

\frac{2 \left(1 - \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{2 \csc{\left(x \right)}}\right)}{\left(1 + \frac{1}{\csc{\left(x \right)}}\right) \left(2 + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}\right) \csc{\left(x \right)}}

    /       2    \          
    |    sin (x) |          
  2*|1 - --------|*sin(x)   
    \    sin(2*x)/          
----------------------------
             /         2   \
             |    2*sin (x)|
(1 + sin(x))*|2 + ---------|
             \     sin(2*x)/

\frac{2 \left(- \frac{\sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 1\right) \sin{\left(x \right)}}{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 2\right) \left(\sin{\left(x \right)} + 1\right)}

      2/x\    4/x\ /    tan(x)\    /x\
16*cot |-|*sin |-|*|1 - ------|*cot|-|
       \4/     \4/ \      2   /    \2/
--------------------------------------
      (1 + sin(x))*(2 + tan(x))

\frac{16 \left(1 - \frac{\tan{\left(x \right)}}{2}\right) \sin^{4}{\left(\frac{x}{4} \right)} \cot^{2}{\left(\frac{x}{4} \right)} \cot{\left(\frac{x}{2} \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\tan{\left(x \right)} + 2\right)}

  /       /    pi\\                   
  |    cos|x - --||                   
  |       \    2 /|    /x\    /x   pi\
4*|1 - -----------|*cos|-|*cos|- - --|
  \      2*cos(x) /    \2/    \2   2 /
--------------------------------------
                   /       /    pi\\  
                   |    cos|x - --||  
 /       /    pi\\ |       \    2 /|  
 |1 + cos|x - --||*|2 + -----------|  
 \       \    2 // \       cos(x)  /

\frac{4 \left(1 - \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{2 \cos{\left(x \right)}}\right) \cos{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(2 + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}\right) \left(\cos{\left(x - \frac{\pi}{2} \right)} + 1\right)}

     2/x\ /    tan(x)\    /x\
4*sin |-|*|1 - ------|*cot|-|
      \2/ \      2   /    \2/
-----------------------------
  (1 + sin(x))*(2 + tan(x))

\frac{4 \left(1 - \frac{\tan{\left(x \right)}}{2}\right) \sin^{2}{\left(\frac{x}{2} \right)} \cot{\left(\frac{x}{2} \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\tan{\left(x \right)} + 2\right)}

  /    tan(x)\    /x\    /x\
4*|1 - ------|*cos|-|*sin|-|
  \      2   /    \2/    \2/
----------------------------
 (1 + sin(x))*(2 + tan(x))

\frac{4 \left(1 - \frac{\tan{\left(x \right)}}{2}\right) \sin{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} \right)}}{\left(\sin{\left(x \right)} + 1\right) \left(\tan{\left(x \right)} + 2\right)}

4*(1 - tan(x)/2)*cos(x/2)*sin(x/2)/((1 + sin(x))*(2 + tan(x)))

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