Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
cos(x)/(1+tan(x))+(-1-tan(x)^2)*sin(x)/(1+tan(x))^2
¿Cómo vas a descomponer esta cos(x)/(1+tan(x))+(-1-tan(x)^2)*sin(x)/(1+tan(x))^2 expresión en fracciones?
Solución
Ha introducido
[src]
/ 2 \
cos(x) \-1 - tan (x)/*sin(x)
---------- + ---------------------
1 + tan(x) 2
(1 + tan(x))
$$\frac{\left(- \tan^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}} + \frac{\cos{\left(x \right)}}{\tan{\left(x \right)} + 1}$$
cos(x)/(1 + tan(x)) + ((-1 - tan(x)^2)*sin(x))/(1 + tan(x))^2
Simplificación general
[src]
___ / pi\ sin(x)
\/ 2 *sin|x + --| - -------
\ 4 / 2
cos (x)
---------------------------
2
(1 + tan(x))
$$\frac{- \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}}$$
(sqrt(2)*sin(x + pi/4) - sin(x)/cos(x)^2)/(1 + tan(x))^2
Unión de expresiones racionales
[src]
/ 2 \
(1 + tan(x))*cos(x) + \-1 - tan (x)/*sin(x)
-------------------------------------------
2
(1 + tan(x))
$$\frac{\left(\tan{\left(x \right)} + 1\right) \cos{\left(x \right)} + \left(- \tan^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}}$$
((1 + tan(x))*cos(x) + (-1 - tan(x)^2)*sin(x))/(1 + tan(x))^2
Respuesta numérica
[src]
cos(x)/(1.0 + tan(x)) + (-1.0 - tan(x)^2)*sin(x)/(1.0 + tan(x))^2
cos(x)/(1.0 + tan(x)) + (-1.0 - tan(x)^2)*sin(x)/(1.0 + tan(x))^2
Denominador común
[src]
cos(x)*tan(x) + 2*sin(x)*tan(x) + cos(x)
-sin(x) + ----------------------------------------
2
1 + tan (x) + 2*tan(x)
$$\frac{2 \sin{\left(x \right)} \tan{\left(x \right)} + \cos{\left(x \right)} \tan{\left(x \right)} + \cos{\left(x \right)}}{\tan^{2}{\left(x \right)} + 2 \tan{\left(x \right)} + 1} - \sin{\left(x \right)}$$
-sin(x) + (cos(x)*tan(x) + 2*sin(x)*tan(x) + cos(x))/(1 + tan(x)^2 + 2*tan(x))
Combinatoria
[src]
2
-sin(x) + cos(x)*tan(x) - tan (x)*sin(x) + cos(x)
-------------------------------------------------
2
(1 + tan(x))
$$\frac{- \sin{\left(x \right)} \tan^{2}{\left(x \right)} - \sin{\left(x \right)} + \cos{\left(x \right)} \tan{\left(x \right)} + \cos{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}}$$
(-sin(x) + cos(x)*tan(x) - tan(x)^2*sin(x) + cos(x))/(1 + tan(x))^2
Denominador racional
[src]
2 / 2 \
(1 + tan(x)) *cos(x) + (1 + tan(x))*\-1 - tan (x)/*sin(x)
---------------------------------------------------------
3
(1 + tan(x))
$$\frac{\left(\tan{\left(x \right)} + 1\right)^{2} \cos{\left(x \right)} + \left(\tan{\left(x \right)} + 1\right) \left(- \tan^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{3}}$$
((1 + tan(x))^2*cos(x) + (1 + tan(x))*(-1 - tan(x)^2)*sin(x))/(1 + tan(x))^3
Compilar la expresión
[src]
/ 2 \
cos(x) \-1 - tan (x)/*sin(x)
---------- + ---------------------
1 + tan(x) 2
(1 + tan(x))
$$\frac{\cos{\left(x \right)}}{\tan{\left(x \right)} + 1} + \frac{\left(- \tan^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}}$$
cos(x)/(1 + tan(x)) + (-1 - tan(x)^2)*sin(x)/(1 + tan(x))^2
Potencias
[src]
/ 2 \
cos(x) \-1 - tan (x)/*sin(x)
---------- + ---------------------
1 + tan(x) 2
(1 + tan(x))
$$\frac{\cos{\left(x \right)}}{\tan{\left(x \right)} + 1} + \frac{\left(- \tan^{2}{\left(x \right)} - 1\right) \sin{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}}$$
/ 2\
| / I*x -I*x\ |
| \- e + e / | / -I*x I*x\
I*x -I*x I*|-1 + -----------------|*\- e + e /
e e | 2 |
---- + ----- | / I*x -I*x\ |
2 2 \ \e + e / /
---------------------- - -------------------------------------------
/ I*x -I*x\ 2
I*\- e + e / / / I*x -I*x\\
1 + ------------------ | I*\- e + e /|
I*x -I*x 2*|1 + ------------------|
e + e | I*x -I*x |
\ e + e /
$$- \frac{i \left(\frac{\left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} - 1\right) \left(e^{i x} - e^{- i x}\right)}{2 \left(\frac{i \left(- e^{i x} + e^{- i x}\right)}{e^{i x} + e^{- i x}} + 1\right)^{2}} + \frac{\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}}{\frac{i \left(- e^{i x} + e^{- i x}\right)}{e^{i x} + e^{- i x}} + 1}$$
(exp(i*x)/2 + exp(-i*x)/2)/(1 + i*(-exp(i*x) + exp(-i*x))/(exp(i*x) + exp(-i*x))) - i*(-1 + (-exp(i*x) + exp(-i*x))^2/(exp(i*x) + exp(-i*x))^2)*(-exp(-i*x) + exp(i*x))/(2*(1 + i*(-exp(i*x) + exp(-i*x))/(exp(i*x) + exp(-i*x)))^2)
Parte trigonométrica
[src]
/ pi\
sin|x + --|
\ 2 / sin(x)
------------- - -----------------------------
2 2
2*sin (x) / 2 \
1 + --------- | 2*sin (x)| 2/ pi\
sin(2*x) |1 + ---------| *sin |x + --|
\ sin(2*x)/ \ 2 /
$$\frac{\sin{\left(x + \frac{\pi}{2} \right)}}{\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 1} - \frac{\sin{\left(x \right)}}{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 1\right)^{2} \sin^{2}{\left(x + \frac{\pi}{2} \right)}}$$
2/x\
8*cot |-|
\4/
1 - --------------
2
/ 2/x\\ /x\
|1 + cot |-|| 4*cot|-|
\ \4// \2/
------------------ - ----------------------------------------------
1 2 / 2 \
1 + ------ / 1 \ / 2/x\\ | -1 + cot (x)|
cot(x) |1 + ------| *|1 + cot |-||*|1 + ------------|
\ cot(x)/ \ \2// | 2 |
\ 1 + cot (x) /
$$\frac{1 - \frac{8 \cot^{2}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}}{1 + \frac{1}{\cot{\left(x \right)}}} - \frac{4 \cot{\left(\frac{x}{2} \right)}}{\left(1 + \frac{1}{\cot{\left(x \right)}}\right)^{2} \left(\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
2/x\
8*tan |-|
\4/
1 - --------------
2
/ 2/x\\ /x\
|1 + tan |-|| 4*tan|-|
\ \4// \2/
------------------ - ---------------------------------------------
1 + tan(x) / 2 \
/ 2/x\\ | 1 - tan (x)| 2
|1 + tan |-||*|1 + -----------|*(1 + tan(x))
\ \2// | 2 |
\ 1 + tan (x)/
$$\frac{1 - \frac{8 \tan^{2}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}}{\tan{\left(x \right)} + 1} - \frac{4 \tan{\left(\frac{x}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\tan{\left(x \right)} + 1\right)^{2}}$$
2/x\ /x\
1 - 2*sin |-| 2*tan|-|
\2/ \2/
------------- - ------------------------------------
1 + tan(x) 2
/ 2/x\\ 2 2/x\
|1 - tan |-|| *(1 + tan(x)) *cos |-|
\ \2// \2/
$$\frac{1 - 2 \sin^{2}{\left(\frac{x}{2} \right)}}{\tan{\left(x \right)} + 1} - \frac{2 \tan{\left(\frac{x}{2} \right)}}{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2} \left(\tan{\left(x \right)} + 1\right)^{2} \cos^{2}{\left(\frac{x}{2} \right)}}$$
1 ___ 2 / pi\
------ - \/ 2 *tan (x)*sin|x + --|
cos(x) \ 4 /
----------------------------------
2
(1 + tan(x))
$$\frac{- \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)} \tan^{2}{\left(x \right)} + \frac{1}{\cos{\left(x \right)}}}{\left(\tan{\left(x \right)} + 1\right)^{2}}$$
cos(x) sin(x)
---------- - ---------------------
1 + tan(x) 2 2
(1 + tan(x)) *cos (x)
$$\frac{\cos{\left(x \right)}}{\tan{\left(x \right)} + 1} - \frac{\sin{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2} \cos^{2}{\left(x \right)}}$$
/ pi\
cos|x - --|
cos(x) \ 2 /
--------------- - --------------------------
/ pi\ 2
cos|x - --| / / pi\\
\ 2 / | cos|x - --||
1 + ----------- | \ 2 /| 2
cos(x) |1 + -----------| *cos (x)
\ cos(x) /
$$\frac{\cos{\left(x \right)}}{1 + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}} - \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}\right)^{2} \cos^{2}{\left(x \right)}}$$
2/x\ /x\
1 - 2*sin |-| 2*cot|-|
\2/ \2/
------------- - -------------------------------------
1 + tan(x) 2
2 / 2/x\\ 2/x\
(1 + tan(x)) *|-1 + cot |-|| *sin |-|
\ \2// \2/
$$\frac{1 - 2 \sin^{2}{\left(\frac{x}{2} \right)}}{\tan{\left(x \right)} + 1} - \frac{2 \cot{\left(\frac{x}{2} \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2} \sin^{2}{\left(\frac{x}{2} \right)}}$$
2/x\
1 - 2*sin |-|
\2/ 2*sin(x)
------------- - ----------------------------
1 + tan(x) 2
(1 + cos(2*x))*(1 + tan(x))
$$\frac{1 - 2 \sin^{2}{\left(\frac{x}{2} \right)}}{\tan{\left(x \right)} + 1} - \frac{2 \sin{\left(x \right)}}{\left(\cos{\left(2 x \right)} + 1\right) \left(\tan{\left(x \right)} + 1\right)^{2}}$$
2
cos(x) sec (x)*sin(x)
---------- - --------------
1 + tan(x) 2
(1 + tan(x))
$$\frac{\cos{\left(x \right)}}{\tan{\left(x \right)} + 1} - \frac{\sin{\left(x \right)} \sec^{2}{\left(x \right)}}{\left(\tan{\left(x \right)} + 1\right)^{2}}$$
2/x\ / 2 \ /x\
1 - tan |-| 2*\-1 - tan (x)/*tan|-|
\2/ \2/
-------------------------- + ---------------------------
/ 2/x\\ / 2/x\\ 2
|1 + tan |-||*(1 + tan(x)) |1 + tan |-||*(1 + tan(x))
\ \2// \ \2//
$$\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\tan{\left(x \right)} + 1\right)} + \frac{2 \left(- \tan^{2}{\left(x \right)} - 1\right) \tan{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\tan{\left(x \right)} + 1\right)^{2}}$$
2/x\ / 2/x\\ /x\
-1 + cot |-| 2*|1 + cot |-||*cot|-|
\2/ \ \2// \2/
-------------------------- - -----------------------------
/ 1 \ / 2/x\\ 2 2
|1 + ------|*|1 + cot |-|| / 1 \ / 2/x\\
\ cot(x)/ \ \2// |1 + ------| *|-1 + cot |-||
\ cot(x)/ \ \2//
$$\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\left(1 + \frac{1}{\cot{\left(x \right)}}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} - \frac{2 \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \cot{\left(\frac{x}{2} \right)}}{\left(1 + \frac{1}{\cot{\left(x \right)}}\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}$$
2
1 - ------------
2/x pi\
sec |- - --|
\2 2 / 2
---------------- - ---------------------------------------------
sec(x) 2
1 + ----------- / 1 \ / sec(x) \ / pi\
/ pi\ |1 + --------|*|1 + -----------| *sec|x - --|
sec|x - --| \ sec(2*x)/ | / pi\| \ 2 /
\ 2 / | sec|x - --||
\ \ 2 //
$$\frac{1 - \frac{2}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}}{\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + 1} - \frac{2}{\left(1 + \frac{1}{\sec{\left(2 x \right)}}\right) \left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec{\left(x - \frac{\pi}{2} \right)}}$$
/ 2 \
| sin (x)|
|-1 - -------|*sin(x)
| 2 |
cos(x) \ cos (x)/
---------- + ---------------------
sin(x) 2
1 + ------ / sin(x)\
cos(x) |1 + ------|
\ cos(x)/
$$\frac{\cos{\left(x \right)}}{\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + 1} + \frac{\left(- \frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - 1\right) \sin{\left(x \right)}}{\left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + 1\right)^{2}}$$
2
1 - -------
2/x\
csc |-|
\2/ 2
--------------- - ---------------------------------------------
/pi \ 2
csc|-- - x| / /pi \\
\2 / | csc|-- - x||
1 + ----------- / 1 \ | \2 /|
csc(x) |1 + -------------|*|1 + -----------| *csc(x)
| /pi \| \ csc(x) /
| csc|-- - 2*x||
\ \2 //
$$\frac{1 - \frac{2}{\csc^{2}{\left(\frac{x}{2} \right)}}}{1 + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}} - \frac{2}{\left(1 + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}\right)^{2} \left(1 + \frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}\right) \csc{\left(x \right)}}$$
2
1 sec (x)
------------------------ - ------------------------------
/ sec(x) \ 2
|1 + -----------|*sec(x) / sec(x) \ / pi\
| / pi\| |1 + -----------| *sec|x - --|
| sec|x - --|| | / pi\| \ 2 /
\ \ 2 // | sec|x - --||
\ \ 2 //
$$\frac{1}{\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(x \right)}} - \frac{\sec^{2}{\left(x \right)}}{\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec{\left(x - \frac{\pi}{2} \right)}}$$
2/pi \
csc |-- - x|
1 \2 /
----------------------------- - -------------------------
/ /pi \\ 2
| csc|-- - x|| / /pi \\
| \2 /| /pi \ | csc|-- - x||
|1 + -----------|*csc|-- - x| | \2 /|
\ csc(x) / \2 / |1 + -----------| *csc(x)
\ csc(x) /
$$\frac{1}{\left(1 + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}\right) \csc{\left(- x + \frac{\pi}{2} \right)}} - \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}\right)^{2} \csc{\left(x \right)}}$$
/ 2/ pi\\
| cos |x - --||
| \ 2 /| / pi\
|-1 - ------------|*cos|x - --|
| 2 | \ 2 /
cos(x) \ cos (x) /
--------------- + -------------------------------
/ pi\ 2
cos|x - --| / / pi\\
\ 2 / | cos|x - --||
1 + ----------- | \ 2 /|
cos(x) |1 + -----------|
\ cos(x) /
$$\frac{\left(-1 - \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right) \cos{\left(x - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}\right)^{2}} + \frac{\cos{\left(x \right)}}{1 + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}}$$
2/pi \
csc |-- - x|
\2 /
-1 - ------------
2
1 csc (x)
----------------------------- + -------------------------
/ /pi \\ 2
| csc|-- - x|| / /pi \\
| \2 /| /pi \ | csc|-- - x||
|1 + -----------|*csc|-- - x| | \2 /|
\ csc(x) / \2 / |1 + -----------| *csc(x)
\ csc(x) /
$$\frac{-1 - \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}}{\left(1 + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}\right)^{2} \csc{\left(x \right)}} + \frac{1}{\left(1 + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}\right) \csc{\left(- x + \frac{\pi}{2} \right)}}$$
/ 4 \
| 4*sin (x)|
|-1 - ---------|*sin(x)
| 2 |
cos(x) \ sin (2*x)/
------------- + -----------------------
2 2
2*sin (x) / 2 \
1 + --------- | 2*sin (x)|
sin(2*x) |1 + ---------|
\ sin(2*x)/
$$\frac{\cos{\left(x \right)}}{\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 1} + \frac{\left(- \frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} - 1\right) \sin{\left(x \right)}}{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 1\right)^{2}}$$
2
sec (x)
-1 - ------------
2/ pi\
sec |x - --|
1 \ 2 /
------------------------ + ------------------------------
/ sec(x) \ 2
|1 + -----------|*sec(x) / sec(x) \ / pi\
| / pi\| |1 + -----------| *sec|x - --|
| sec|x - --|| | / pi\| \ 2 /
\ \ 2 // | sec|x - --||
\ \ 2 //
$$\frac{1}{\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(x \right)}} + \frac{- \frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} - 1}{\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec{\left(x - \frac{\pi}{2} \right)}}$$
2/x\ / 2/x\\ /x\
1 - tan |-| 2*|1 + tan |-||*tan|-|
\2/ \ \2// \2/
-------------------------- - ----------------------------
/ 2/x\\ 2
|1 + tan |-||*(1 + tan(x)) / 2/x\\ 2
\ \2// |1 - tan |-|| *(1 + tan(x))
\ \2//
$$\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\tan{\left(x \right)} + 1\right)} - \frac{2 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}}{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2} \left(\tan{\left(x \right)} + 1\right)^{2}}$$
2
sec (x)
-1 - -------
2
1 csc (x)
------------------- + --------------------
/ sec(x)\ 2
|1 + ------|*sec(x) / sec(x)\
\ csc(x)/ |1 + ------| *csc(x)
\ csc(x)/
$$\frac{-1 - \frac{\sec^{2}{\left(x \right)}}{\csc^{2}{\left(x \right)}}}{\left(1 + \frac{\sec{\left(x \right)}}{\csc{\left(x \right)}}\right)^{2} \csc{\left(x \right)}} + \frac{1}{\left(1 + \frac{\sec{\left(x \right)}}{\csc{\left(x \right)}}\right) \sec{\left(x \right)}}$$
2/x\
1 - 2*sin |-|
\2/ 2*sin(x)
------------- - ------------------------------------
2 2
2*sin (x) / 2 \
1 + --------- | 2*sin (x)| / /pi \\
sin(2*x) |1 + ---------| *|1 + sin|-- + 2*x||
\ sin(2*x)/ \ \2 //
$$\frac{1 - 2 \sin^{2}{\left(\frac{x}{2} \right)}}{\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 1} - \frac{2 \sin{\left(x \right)}}{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 1\right)^{2} \left(\sin{\left(2 x + \frac{\pi}{2} \right)} + 1\right)}$$
/ 4 \
| 4*sin (x)|
/ pi\ |-1 - ---------|*sin(x)
sin|x + --| | 2 |
\ 2 / \ sin (2*x)/
------------- + -----------------------
2 2
2*sin (x) / 2 \
1 + --------- | 2*sin (x)|
sin(2*x) |1 + ---------|
\ sin(2*x)/
$$\frac{\sin{\left(x + \frac{\pi}{2} \right)}}{\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 1} + \frac{\left(- \frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} - 1\right) \sin{\left(x \right)}}{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 1\right)^{2}}$$
2/x pi\ / pi\
1 - 2*cos |- - --| 2*cos|x - --|
\2 2 / \ 2 /
------------------ - ---------------------------------
/ pi\ 2
cos|x - --| / / pi\\
\ 2 / | cos|x - --||
1 + ----------- | \ 2 /|
cos(x) |1 + -----------| *(1 + cos(2*x))
\ cos(x) /
$$\frac{1 - 2 \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{1 + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}} - \frac{2 \cos{\left(x - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}\right)^{2} \left(\cos{\left(2 x \right)} + 1\right)}$$
/ 1 \ /x\
2/x\ 2*|-1 - -------|*cot|-|
-1 + cot |-| | 2 | \2/
\2/ \ cot (x)/
-------------------------- + ---------------------------
/ 1 \ / 2/x\\ 2
|1 + ------|*|1 + cot |-|| / 1 \ / 2/x\\
\ cot(x)/ \ \2// |1 + ------| *|1 + cot |-||
\ cot(x)/ \ \2//
$$\frac{2 \left(-1 - \frac{1}{\cot^{2}{\left(x \right)}}\right) \cot{\left(\frac{x}{2} \right)}}{\left(1 + \frac{1}{\cot{\left(x \right)}}\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\left(1 + \frac{1}{\cot{\left(x \right)}}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
(-1 + cot(x/2)^2)/((1 + 1/cot(x))*(1 + cot(x/2)^2)) + 2*(-1 - 1/cot(x)^2)*cot(x/2)/((1 + 1/cot(x))^2*(1 + cot(x/2)^2))