Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
cos((\pi)/(2)+a)cos(\pi-a)+sin((3\pi)/(2)+a)sin(\pi+a)
¿Cómo vas a descomponer esta cos((\pi)/(2)+a)cos(\pi-a)+sin((3\pi)/(2)+a)sin(\pi+a) expresión en fracciones?
Solución
Ha introducido
[src]
//3 \ \
||--| |
/pi \ |\pi/ |
cos|-- + a|*cos(pi - a) + sin|---- + a|*sin(pi + a)
\2 / \ 2 /
$$\sin{\left(a + \pi \right)} \sin{\left(a + \frac{3 \frac{1}{\pi}}{2} \right)} + \cos{\left(\pi - a \right)} \cos{\left(a + \frac{\pi}{2} \right)}$$
cos(pi/2 + a)*cos(pi - a) + sin((3/pi)/2 + a)*sin(pi + a)
Simplificación general
[src]
/ / 3 \ \ |- sin|a + ----| + cos(a)|*sin(a) \ \ 2*pi/ /
$$\left(- \sin{\left(a + \frac{3}{2 \pi} \right)} + \cos{\left(a \right)}\right) \sin{\left(a \right)}$$
(-sin(a + 3/(2*pi)) + cos(a))*sin(a)
Respuesta numérica
[src]
cos(pi - a)*cos(pi/2 + a) + sin(pi + a)*sin((3/pi)/2 + a)
cos(pi - a)*cos(pi/2 + a) + sin(pi + a)*sin((3/pi)/2 + a)
Combinatoria
[src]
/ / 3 \ \ |- sin|a + ----| + cos(a)|*sin(a) \ \ 2*pi/ /
$$\left(- \sin{\left(a + \frac{3}{2 \pi} \right)} + \cos{\left(a \right)}\right) \sin{\left(a \right)}$$
(-sin(a + 3/(2*pi)) + cos(a))*sin(a)
Potencias
[src]
/ / pi\ / pi\\ / / 3 \ / 3 \\
| I*|a + --| I*|-a - --|| | I*|-a - ----| I*|a + ----||
/ I*(pi - a) I*(a - pi)\ | \ 2 / \ 2 /| / I*(-pi - a) I*(pi + a)\ | \ 2*pi/ \ 2*pi/|
|e e | |e e | \- e + e /*\- e + e /
|----------- + -----------|*|----------- + ------------| - -----------------------------------------------------------------
\ 2 2 / \ 2 2 / 4
$$\left(\frac{e^{i \left(\pi - a\right)}}{2} + \frac{e^{i \left(a - \pi\right)}}{2}\right) \left(\frac{e^{i \left(- a - \frac{\pi}{2}\right)}}{2} + \frac{e^{i \left(a + \frac{\pi}{2}\right)}}{2}\right) - \frac{\left(- e^{i \left(- a - \frac{3}{2 \pi}\right)} + e^{i \left(a + \frac{3}{2 \pi}\right)}\right) \left(- e^{i \left(- a - \pi\right)} + e^{i \left(a + \pi\right)}\right)}{4}$$
/ 3 \
cos(a)*sin(a) - sin(a)*sin|a + ----|
\ 2*pi/
$$- \sin{\left(a \right)} \sin{\left(a + \frac{3}{2 \pi} \right)} + \sin{\left(a \right)} \cos{\left(a \right)}$$
cos(a)*sin(a) - sin(a)*sin(a + 3/(2*pi))
Unión de expresiones racionales
[src]
/pi + 2*a\ /3 + 2*pi*a\
- cos(a)*cos|--------| - sin(a)*sin|----------|
\ 2 / \ 2*pi /
$$- \sin{\left(a \right)} \sin{\left(\frac{2 \pi a + 3}{2 \pi} \right)} - \cos{\left(a \right)} \cos{\left(\frac{2 a + \pi}{2} \right)}$$
-cos(a)*cos((pi + 2*a)/2) - sin(a)*sin((3 + 2*pi*a)/(2*pi))
Denominador racional
[src]
/ pi\ / 3 \
- cos(a)*cos|a + --| - sin(a)*sin|a + ----|
\ 2 / \ 2*pi/
$$- \sin{\left(a \right)} \sin{\left(a + \frac{3}{2 \pi} \right)} - \cos{\left(a \right)} \cos{\left(a + \frac{\pi}{2} \right)}$$
-cos(a)*cos(a + pi/2) - sin(a)*sin(a + 3/(2*pi))
Denominador común
[src]
/ 3 \
sin(pi + a)*sin|a + ----| - cos(pi - a)*sin(a)
\ 2*pi/
$$- \sin{\left(a \right)} \cos{\left(\pi - a \right)} + \sin{\left(a + \frac{3}{2 \pi} \right)} \sin{\left(a + \pi \right)}$$
sin(pi + a)*sin(a + 3/(2*pi)) - cos(pi - a)*sin(a)
Parte trigonométrica
[src]
/ 2 \
2*|-1 + -------|
| 2/a\| /a 3 \
| sec |-|| 4*sec|- + ----|
\ \2// \2 4*pi/
------------------ - ---------------------------------------------------------------
/a\ /a pi\ / 2/a 3 \ \
sec|-|*sec|- - --| | sec |- + ----| |
\2/ \2 2 / | \2 4*pi/ | /a\ /a pi\ /a pi 3 \
|1 + -------------------|*sec|-|*sec|- - --|*sec|- - -- + ----|
| 2/a pi 3 \| \2/ \2 2 / \2 2 4*pi/
| sec |- - -- + ----||
\ \2 2 4*pi//
$$\frac{2 \left(-1 + \frac{2}{\sec^{2}{\left(\frac{a}{2} \right)}}\right)}{\sec{\left(\frac{a}{2} \right)} \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - \frac{4 \sec{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} + \frac{3}{4 \pi} \right)}} + 1\right) \sec{\left(\frac{a}{2} \right)} \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)} \sec{\left(\frac{a}{2} - \frac{\pi}{2} + \frac{3}{4 \pi} \right)}}$$
/ 2/a\ \ / 2/a pi\\
| sec |-| | | sec |- - --||
/ 1 \ | \2/ | | \2 4 /|
|1 + -----------|*|-1 + ------------|*|-1 + ------------|
| / pi\| | 2/a pi\| | 2/a pi\|
| sec|a - --|| | sec |- - --|| | sec |- + --||
1 1 \ \ 2 // \ \2 2 // \ \2 4 //
----------------- - ----------- + ---------------------------------------------------------
/ 3 \ / 3 \ 2/a\
2*sec|2*a + ----| 2*sec|----| 2*sec |-|
\ 2*pi/ \2*pi/ \2/
$$\frac{\left(1 + \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}\right) \left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1\right) \left(\frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}{\sec^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} - 1\right)}{2 \sec^{2}{\left(\frac{a}{2} \right)}} - \frac{1}{2 \sec{\left(\frac{3}{2 \pi} \right)}} + \frac{1}{2 \sec{\left(2 a + \frac{3}{2 \pi} \right)}}$$
/a\ /a 3 \ /a\ /a 3 \
4*cos|-|*sec|- + ----|*sin|-|*sin|- + ----|
sin(2*a) \2/ \2 4*pi/ \2/ \2 4*pi/
-------- - -------------------------------------------
2 2/a 3 \
1 + tan |- + ----|
\2 4*pi/
$$\frac{\sin{\left(2 a \right)}}{2} - \frac{4 \sin{\left(\frac{a}{2} \right)} \sin{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} \cos{\left(\frac{a}{2} \right)} \sec{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} + 1}$$
2/a\ 4/a\ /a\ /a 3 \
16*cot |-|*sin |-|*cot|-|*tan|- + ----|
sin(2*a) \4/ \4/ \2/ \2 4*pi/
-------- - ---------------------------------------
2 2/a 3 \
1 + tan |- + ----|
\2 4*pi/
$$\frac{\sin{\left(2 a \right)}}{2} - \frac{16 \sin^{4}{\left(\frac{a}{4} \right)} \tan{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} \cot^{2}{\left(\frac{a}{4} \right)} \cot{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} + 1}$$
/ 1 \ / 2/a pi\\ |1 - -------|*|1 - tan |- + --|| | 2/a\| \ \2 4 // /a 3 \ | tan |-|| 4*tan|- + ----| \ \2// \2 4*pi/ -------------------------------- - ----------------------------------------- / 1 \ / 2/a pi\\ / 1 \ / 2/a 3 \\ /a\ |1 + -------|*|1 + tan |- + --|| |1 + -------|*|1 + tan |- + ----||*tan|-| | 2/a\| \ \2 4 // | 2/a\| \ \2 4*pi// \2/ | tan |-|| | tan |-|| \ \2// \ \2//
$$\frac{\left(1 - \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 - \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}\right)}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)} - \frac{4 \tan{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \left(\tan^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} + 1\right) \tan{\left(\frac{a}{2} \right)}}$$
1 1
------------------ - --------------------
/pi \ / 3 \
sec(a)*sec|-- - a| csc(a)*csc|a + ----|
\2 / \ 2*pi/
$$\frac{1}{\sec{\left(a \right)} \sec{\left(- a + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(a \right)} \csc{\left(a + \frac{3}{2 \pi} \right)}}$$
1 1
------------- - --------------------
csc(a)*sec(a) / 3 \
csc(a)*csc|a + ----|
\ 2*pi/
$$\frac{1}{\csc{\left(a \right)} \sec{\left(a \right)}} - \frac{1}{\csc{\left(a \right)} \csc{\left(a + \frac{3}{2 \pi} \right)}}$$
/ 3 \ / 3 \ 2/a\ / 2/a pi\\ / 2/a\\
- cos|2*a + ----| + cos|----| cos |-|*(1 + sin(a))*|-1 + cot |- + --||*|-1 + tan |-||
\ 2*pi/ \2*pi/ \2/ \ \2 4 // \ \2//
- ----------------------------- + -------------------------------------------------------
2 2
$$\frac{\left(\sin{\left(a \right)} + 1\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} - 1\right) \left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right) \cos^{2}{\left(\frac{a}{2} \right)}}{2} - \frac{- \cos{\left(2 a + \frac{3}{2 \pi} \right)} + \cos{\left(\frac{3}{2 \pi} \right)}}{2}$$
/ 2 \
2*|-1 + ------------|
| 2/pi a\| /pi 3 a\
| csc |-- - -|| 4*csc|-- - ---- - -|
\ \2 2// \2 4*pi 2/
--------------------- - ----------------------------------------------------------
/a\ /pi a\ / 2/pi 3 a\\
csc|-|*csc|-- - -| | csc |-- - ---- - -||
\2/ \2 2/ | \2 4*pi 2/| /a\ /pi a\ /a 3 \
|1 + -------------------|*csc|-|*csc|-- - -|*csc|- + ----|
| 2/a 3 \ | \2/ \2 2/ \2 4*pi/
| csc |- + ----| |
\ \2 4*pi/ /
$$\frac{2 \left(-1 + \frac{2}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}\right)}{\csc{\left(\frac{a}{2} \right)} \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - \frac{4 \csc{\left(- \frac{a}{2} - \frac{3}{4 \pi} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} - \frac{3}{4 \pi} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}\right) \csc{\left(\frac{a}{2} \right)} \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)} \csc{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}$$
/a\ /a pi\ /a pi 3 \
4*cos|-|*cos|- - --|*cos|- - -- + ----|
/ 2/a\\ /a\ /a pi\ \2/ \2 2 / \2 2 4*pi/
2*|-1 + 2*cos |-||*cos|-|*cos|- - --| - ---------------------------------------
\ \2// \2/ \2 2 / / 2/a pi 3 \\
| cos |- - -- + ----||
| \2 2 4*pi/| /a 3 \
|1 + -------------------|*cos|- + ----|
| 2/a 3 \ | \2 4*pi/
| cos |- + ----| |
\ \2 4*pi/ /
$$2 \left(2 \cos^{2}{\left(\frac{a}{2} \right)} - 1\right) \cos{\left(\frac{a}{2} \right)} \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)} - \frac{4 \cos{\left(\frac{a}{2} \right)} \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)} \cos{\left(\frac{a}{2} - \frac{\pi}{2} + \frac{3}{4 \pi} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} + \frac{3}{4 \pi} \right)}}{\cos^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}\right) \cos{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}$$
1 1
------ - ------------------
sec(a) / pi 3 \
sec|a - -- + ----|
\ 2 2*pi/
---------------------------
/ pi\
sec|a - --|
\ 2 /
$$\frac{- \frac{1}{\sec{\left(a - \frac{\pi}{2} + \frac{3}{2 \pi} \right)}} + \frac{1}{\sec{\left(a \right)}}}{\sec{\left(a - \frac{\pi}{2} \right)}}$$
/ 3 \
cos(a)*sin(a) - sin(a)*sin|a + ----|
\ 2*pi/
$$- \sin{\left(a \right)} \sin{\left(a + \frac{3}{2 \pi} \right)} + \sin{\left(a \right)} \cos{\left(a \right)}$$
/ pi\ / 3 \
cos(a)*cos|a - --| - sin(a)*sin|a + ----|
\ 2 / \ 2*pi/
$$- \sin{\left(a \right)} \sin{\left(a + \frac{3}{2 \pi} \right)} + \cos{\left(a \right)} \cos{\left(a - \frac{\pi}{2} \right)}$$
/ / pi 3 \ \ / pi\ |- cos|a - -- + ----| + cos(a)|*cos|a - --| \ \ 2 2*pi/ / \ 2 /
$$\left(\cos{\left(a \right)} - \cos{\left(a - \frac{\pi}{2} + \frac{3}{2 \pi} \right)}\right) \cos{\left(a - \frac{\pi}{2} \right)}$$
/ 3 \ / 3 \
cos|2*a + ----| cos|----|
\ 2*pi/ sin(2*a) \2*pi/
--------------- + -------- - ---------
2 2 2
$$\frac{\sin{\left(2 a \right)}}{2} + \frac{\cos{\left(2 a + \frac{3}{2 \pi} \right)}}{2} - \frac{\cos{\left(\frac{3}{2 \pi} \right)}}{2}$$
/ pi\ / pi 3 \
cos(a)*sin(a) + cos|a + --|*cos|a - -- + ----|
\ 2 / \ 2 2*pi/
$$\sin{\left(a \right)} \cos{\left(a \right)} + \cos{\left(a + \frac{\pi}{2} \right)} \cos{\left(a - \frac{\pi}{2} + \frac{3}{2 \pi} \right)}$$
/ 1 \ / 2/a pi\\ |-1 + -------|*|-1 + cot |- + --|| | 2/a\| \ \2 4 // /a 3 \ | cot |-|| 4*cot|- + ----| \ \2// \2 4*pi/ ---------------------------------- - ----------------------------------------- / 1 \ / 2/a pi\\ / 1 \ / 2/a 3 \\ /a\ |1 + -------|*|1 + cot |- + --|| |1 + -------|*|1 + cot |- + ----||*cot|-| | 2/a\| \ \2 4 // | 2/a\| \ \2 4*pi// \2/ | cot |-|| | cot |-|| \ \2// \ \2//
$$\frac{\left(-1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)} - \frac{4 \cot{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \left(\cot^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} + 1\right) \cot{\left(\frac{a}{2} \right)}}$$
/ 2/a\ /a 3 \ \
|1 - tan |-| 2*tan|- + ----| |
| \2/ \2 4*pi/ | /a\
2*|----------- - ------------------|*tan|-|
| 2/a\ 2/a 3 \| \2/
|1 + tan |-| 1 + tan |- + ----||
\ \2/ \2 4*pi//
-------------------------------------------
2/a\
1 + tan |-|
\2/
$$\frac{2 \left(\frac{1 - \tan^{2}{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} - \frac{2 \tan{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} + 1}\right) \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}$$
1 1
------------------ - --------------------
/pi \ / 3 \
csc(a)*csc|-- - a| csc(a)*csc|a + ----|
\2 / \ 2*pi/
$$- \frac{1}{\csc{\left(a \right)} \csc{\left(a + \frac{3}{2 \pi} \right)}} + \frac{1}{\csc{\left(a \right)} \csc{\left(- a + \frac{\pi}{2} \right)}}$$
/ 2/a pi\\ / 2/a\\ /a 3 \ /a\ |-1 + cot |- + --||*|-1 + tan |-|| 4*cot|- + ----|*tan|-| \ \2 4 // \ \2// \2 4*pi/ \2/ ---------------------------------- - ---------------------------------- / 2/a pi\\ / 2/a\\ / 2/a 3 \\ / 2/a\\ |1 + cot |- + --||*|1 + tan |-|| |1 + cot |- + ----||*|1 + tan |-|| \ \2 4 // \ \2// \ \2 4*pi// \ \2//
$$\frac{\left(\tan^{2}{\left(\frac{a}{2} \right)} - 1\right) \left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)} - \frac{4 \tan{\left(\frac{a}{2} \right)} \cot{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} + 1\right)}$$
2/a 3 \
4*sin |- + ----|*sin(a)
sin(2*a) \2 4*pi/
-------- - ------------------------------------
2 / 4/a 3 \\
| 4*sin |- + ----||
| \2 4*pi/| / 3 \
|1 + ----------------|*sin|a + ----|
| 2/ 3 \ | \ 2*pi/
| sin |a + ----| |
\ \ 2*pi/ /
$$\frac{\sin{\left(2 a \right)}}{2} - \frac{4 \sin{\left(a \right)} \sin^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\sin^{2}{\left(a + \frac{3}{2 \pi} \right)}} + 1\right) \sin{\left(a + \frac{3}{2 \pi} \right)}}$$
/ 2/a\\ / 2/a pi\\ /a\ /a 3 \ |1 - cot |-||*|1 - tan |- + --|| 4*cot|-|*tan|- + ----| \ \2// \ \2 4 // \2/ \2 4*pi/ -------------------------------- - ---------------------------------- / 2/a\\ / 2/a pi\\ / 2/a\\ / 2/a 3 \\ |1 + cot |-||*|1 + tan |- + --|| |1 + cot |-||*|1 + tan |- + ----|| \ \2// \ \2 4 // \ \2// \ \2 4*pi//
$$\frac{\left(1 - \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}\right) \left(1 - \cot^{2}{\left(\frac{a}{2} \right)}\right)}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)} - \frac{4 \tan{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} \cot{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
/ / 3 \\ /a\ /a\ /a 3 \
2*|1 + cos|a + ----||*cos|-|*sin|-|*sin|- + ----|
sin(2*a) \ \ 2*pi// \2/ \2/ \2 4*pi/
-------- - -------------------------------------------------
2 /a 3 \
cos|- + ----|
\2 4*pi/
$$- \frac{2 \left(\cos{\left(a + \frac{3}{2 \pi} \right)} + 1\right) \sin{\left(\frac{a}{2} \right)} \sin{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} \cos{\left(\frac{a}{2} \right)}}{\cos{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}} + \frac{\sin{\left(2 a \right)}}{2}$$
2/a\ /a\ /a 3 \
4*sin |-|*cot|-|*tan|- + ----|
/ 2/a\\ /a\ /a\ \2/ \2/ \2 4*pi/
2*|-1 + 2*cos |-||*cos|-|*sin|-| - ------------------------------
\ \2// \2/ \2/ 2/a 3 \
1 + tan |- + ----|
\2 4*pi/
$$2 \left(2 \cos^{2}{\left(\frac{a}{2} \right)} - 1\right) \sin{\left(\frac{a}{2} \right)} \cos{\left(\frac{a}{2} \right)} - \frac{4 \sin^{2}{\left(\frac{a}{2} \right)} \tan{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} \cot{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} + 1}$$
/ 2\
| / 2/a\\ |
| 2*|1 - tan |-|| |
/ 2/a\\ | \ \4// | /a\
4*|1 - tan |-||*|-1 + ----------------|*tan|-|
\ \4// | 2 | \4/
| / 2/a\\ | 2/a\ /a 3 \
| |1 + tan |-|| | 16*tan |-|*tan|- + ----|
\ \ \4// / \4/ \2 4*pi/
---------------------------------------------- - ------------------------------------------
2 2
/ 2/a\\ / 2/a\\ / 2/a 3 \\ /a\
|1 + tan |-|| |1 + tan |-|| *|1 + tan |- + ----||*tan|-|
\ \4// \ \4// \ \2 4*pi// \2/
$$\frac{4 \left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right) \left(\frac{2 \left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}} - 1\right) \tan{\left(\frac{a}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}} - \frac{16 \tan^{2}{\left(\frac{a}{4} \right)} \tan{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} + 1\right) \tan{\left(\frac{a}{2} \right)}}$$
/ 2/a\ /a 3 \ \
|-1 + cot |-| 2*cot|- + ----| |
| \2/ \2 4*pi/ | /a\
2*|------------ - ------------------|*cot|-|
| 2/a\ 2/a 3 \| \2/
|1 + cot |-| 1 + cot |- + ----||
\ \2/ \2 4*pi//
--------------------------------------------
2/a\
1 + cot |-|
\2/
$$\frac{2 \left(\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} - \frac{2 \cot{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} + 1}\right) \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}$$
/ 2/pi a\\ / 2/a pi\ \
| csc |-- - -|| | csc |- + --| |
/ 1 \ | \2 2/| | \2 4 / |
|1 + ------|*|-1 + ------------|*|-1 + --------------|
\ csc(a)/ | 2/a\ | | 2/ a pi\|
| csc |-| | | csc |- - + --||
1 1 \ \2/ / \ \ 2 4 //
---------------------- - ---------------- + ------------------------------------------------------
/pi 3 \ /pi 3 \ 2/pi a\
2*csc|-- - 2*a - ----| 2*csc|-- - ----| 2*csc |-- - -|
\2 2*pi/ \2 2*pi/ \2 2/
$$\frac{\left(-1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \left(-1 + \frac{\csc^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{4} \right)}}\right) \left(1 + \frac{1}{\csc{\left(a \right)}}\right)}{2 \csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - \frac{1}{2 \csc{\left(- \frac{3}{2 \pi} + \frac{\pi}{2} \right)}} + \frac{1}{2 \csc{\left(- 2 a - \frac{3}{2 \pi} + \frac{\pi}{2} \right)}}$$
/ / 3 \ / pi\\ |- sin|a + ----| + sin|a + --||*sin(a) \ \ 2*pi/ \ 2 //
$$\left(- \sin{\left(a + \frac{3}{2 \pi} \right)} + \sin{\left(a + \frac{\pi}{2} \right)}\right) \sin{\left(a \right)}$$
/ /a\ \
2 | 2*tan|-| |
/ 2/a\\ | \2/ | / 2/a\\ / 1 \
|1 - tan |-|| *|1 + -----------|*|-1 + tan |-||*|-1 + ------------|
2/ 3 \ 2/ 3 \ \ \4// | 2/a\| \ \2// | 2/a pi\|
1 - tan |a + ----| 1 - tan |----| | 1 + tan |-|| | tan |- + --||
\ 4*pi/ \4*pi/ \ \2// \ \2 4 //
---------------------- - ------------------ + -------------------------------------------------------------------
/ 2/ 3 \\ / 2/ 3 \\ 2
2*|1 + tan |a + ----|| 2*|1 + tan |----|| / 2/a\\
\ \ 4*pi// \ \4*pi// 2*|1 + tan |-||
\ \4//
$$\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}\right) \left(1 + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{2} \left(\tan^{2}{\left(\frac{a}{2} \right)} - 1\right)}{2 \left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}} + \frac{1 - \tan^{2}{\left(a + \frac{3}{4 \pi} \right)}}{2 \left(\tan^{2}{\left(a + \frac{3}{4 \pi} \right)} + 1\right)} - \frac{1 - \tan^{2}{\left(\frac{3}{4 \pi} \right)}}{2 \left(\tan^{2}{\left(\frac{3}{4 \pi} \right)} + 1\right)}$$
/ 3 \ / 3 \ 2/a\
cos|2*a + ----| cos|----| 2*cos |-|*cos(a)*sin(a)
\ 2*pi/ \2*pi/ \2/
--------------- - --------- + -----------------------
2 2 1 + cos(a)
$$\frac{\cos{\left(2 a + \frac{3}{2 \pi} \right)}}{2} - \frac{\cos{\left(\frac{3}{2 \pi} \right)}}{2} + \frac{2 \sin{\left(a \right)} \cos^{2}{\left(\frac{a}{2} \right)} \cos{\left(a \right)}}{\cos{\left(a \right)} + 1}$$
4/a\ 2/a\ /a 3 \
16*cos |-|*tan |-|*tan|- + ----|
sin(2*a) \4/ \4/ \2 4*pi/
-------- - --------------------------------
2 / 2/a 3 \\ /a\
|1 + tan |- + ----||*tan|-|
\ \2 4*pi// \2/
$$\frac{\sin{\left(2 a \right)}}{2} - \frac{16 \cos^{4}{\left(\frac{a}{4} \right)} \tan^{2}{\left(\frac{a}{4} \right)} \tan{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} + 1\right) \tan{\left(\frac{a}{2} \right)}}$$
/ 4/a\\
| 4*sin |-|| / 2 \
2/pi a\ | \2/| | cos (a) |
sin |-- + -|*(1 + sin(a))*|-1 + ---------|*|-1 + --------------|
/pi 3 \ /pi 3 \ \2 2/ | 2 | | 4/a pi\|
sin|-- + 2*a + ----| sin|-- + ----| \ sin (a) / | 4*sin |- + --||
\2 2*pi/ \2 2*pi/ \ \2 4 //
-------------------- - -------------- + ----------------------------------------------------------------
2 2 2
$$\frac{\left(-1 + \frac{\cos^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}\right) \left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} - 1\right) \left(\sin{\left(a \right)} + 1\right) \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{2} \right)}}{2} + \frac{\sin{\left(2 a + \frac{3}{2 \pi} + \frac{\pi}{2} \right)}}{2} - \frac{\sin{\left(\frac{3}{2 \pi} + \frac{\pi}{2} \right)}}{2}$$
2/a 3 \
4*sin |- + ----|*sin(a)
/ 2/pi a\\ /a\ /pi a\ \2 4*pi/
2*|-1 + 2*sin |-- + -||*sin|-|*sin|-- + -| - ------------------------------------
\ \2 2// \2/ \2 2/ / 4/a 3 \\
| 4*sin |- + ----||
| \2 4*pi/| / 3 \
|1 + ----------------|*sin|a + ----|
| 2/ 3 \ | \ 2*pi/
| sin |a + ----| |
\ \ 2*pi/ /
$$2 \left(2 \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{2} \right)} - 1\right) \sin{\left(\frac{a}{2} \right)} \sin{\left(\frac{a}{2} + \frac{\pi}{2} \right)} - \frac{4 \sin{\left(a \right)} \sin^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\sin^{2}{\left(a + \frac{3}{2 \pi} \right)}} + 1\right) \sin{\left(a + \frac{3}{2 \pi} \right)}}$$
2
/ 2/a\\ 4/a\ / 2/a\\ / 1 \
|1 - tan |-|| *cos |-|*(1 + sin(a))*|-1 + tan |-||*|-1 + ------------|
2/ 3 \ 2/ 3 \ \ \4// \4/ \ \2// | 2/a pi\|
1 - tan |a + ----| 1 - tan |----| | tan |- + --||
\ 4*pi/ \4*pi/ \ \2 4 //
---------------------- - ------------------ + ----------------------------------------------------------------------
/ 2/ 3 \\ / 2/ 3 \\ 2
2*|1 + tan |a + ----|| 2*|1 + tan |----||
\ \ 4*pi// \ \4*pi//
$$\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}\right) \left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{2} \left(\sin{\left(a \right)} + 1\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} - 1\right) \cos^{4}{\left(\frac{a}{4} \right)}}{2} + \frac{1 - \tan^{2}{\left(a + \frac{3}{4 \pi} \right)}}{2 \left(\tan^{2}{\left(a + \frac{3}{4 \pi} \right)} + 1\right)} - \frac{1 - \tan^{2}{\left(\frac{3}{4 \pi} \right)}}{2 \left(\tan^{2}{\left(\frac{3}{4 \pi} \right)} + 1\right)}$$
/ 3*pi\ / 3 \
sin(pi + a)*sin|-a + ----| - sin(a)*sin|a + ----|
\ 2 / \ 2*pi/
$$- \sin{\left(a \right)} \sin{\left(a + \frac{3}{2 \pi} \right)} + \sin{\left(- a + \frac{3 \pi}{2} \right)} \sin{\left(a + \pi \right)}$$
/ /a\ \
2 | 2*cot|-| |
/ 2/a\\ | \2/ | / 1 \ / 2/a pi\\
|-1 + cot |-|| *|1 + -----------|*|-1 + -------|*|-1 + cot |- + --||
2/ 3 \ 2/ 3 \ \ \4// | 2/a\| | 2/a\| \ \2 4 //
-1 + cot |a + ----| -1 + cot |----| | 1 + cot |-|| | cot |-||
\ 4*pi/ \4*pi/ \ \2// \ \2//
---------------------- - ------------------ + --------------------------------------------------------------------
/ 2/ 3 \\ / 2/ 3 \\ 2
2*|1 + cot |a + ----|| 2*|1 + cot |----|| / 2/a\\
\ \ 4*pi// \ \4*pi// 2*|1 + cot |-||
\ \4//
$$\frac{\left(-1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)}{2 \left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}} + \frac{\cot^{2}{\left(a + \frac{3}{4 \pi} \right)} - 1}{2 \left(\cot^{2}{\left(a + \frac{3}{4 \pi} \right)} + 1\right)} - \frac{-1 + \cot^{2}{\left(\frac{3}{4 \pi} \right)}}{2 \left(1 + \cot^{2}{\left(\frac{3}{4 \pi} \right)}\right)}$$
2
/ 3 \ / 3 \ / 2/a\\ 4/a\ / 2/a pi\\ / 2/a\\
cos|2*a + ----| cos|----| |-1 + cot |-|| *sin |-|*(1 + sin(a))*|-1 + cot |- + --||*|-1 + tan |-||
\ 2*pi/ \2*pi/ \ \4// \4/ \ \2 4 // \ \2//
--------------- - --------- + -----------------------------------------------------------------------
2 2 2
$$\frac{\left(\sin{\left(a \right)} + 1\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} - 1\right) \left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right) \sin^{4}{\left(\frac{a}{4} \right)}}{2} + \frac{\cos{\left(2 a + \frac{3}{2 \pi} \right)}}{2} - \frac{\cos{\left(\frac{3}{2 \pi} \right)}}{2}$$
/ /a 3 \ \
| 2*tan|- + ----| |
2/a\ | \2 4*pi/ | /a\
2*cos |-|*|- ------------------ + cos(a)|*tan|-|
\2/ | 2/a 3 \ | \2/
| 1 + tan |- + ----| |
\ \2 4*pi/ /
$$2 \left(\cos{\left(a \right)} - \frac{2 \tan{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} + 1}\right) \cos^{2}{\left(\frac{a}{2} \right)} \tan{\left(\frac{a}{2} \right)}$$
/ 2\
| / 2/a\\ |
| 2*|-1 + cot |-|| |
/ 2/a\\ | \ \4// | /a\
4*|-1 + cot |-||*|-1 + -----------------|*cot|-|
\ \4// | 2 | \4/
| / 2/a\\ | 2/a\ /a\
| |1 + cot |-|| | 16*cot |-|*cot|-|
\ \ \4// / \4/ \2/
------------------------------------------------ - -------------------------------------------------
2 2
/ 2/a\\ / 2/a\\ / 1 \ /a 3 \
|1 + cot |-|| |1 + cot |-|| *|1 + --------------|*cot|- + ----|
\ \4// \ \4// | 2/a 3 \| \2 4*pi/
| cot |- + ----||
\ \2 4*pi//
$$\frac{4 \left(\frac{2 \left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}} - 1\right) \left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right) \cot{\left(\frac{a}{4} \right)}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}} - \frac{16 \cot^{2}{\left(\frac{a}{4} \right)} \cot{\left(\frac{a}{2} \right)}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}\right) \left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2} \cot{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}$$
1 1
----------- - -------------
/pi \ / 3 \
csc|-- - a| csc|a + ----|
\2 / \ 2*pi/
---------------------------
csc(a)
$$\frac{- \frac{1}{\csc{\left(a + \frac{3}{2 \pi} \right)}} + \frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}}}{\csc{\left(a \right)}}$$
/ 2/a pi\\ / 2/a pi\\
| cos |- - --|| | cos |- + --||
2/a\ / / pi\\ | \2 2 /| | \2 4 /|
cos |-|*|1 + cos|a - --||*|-1 + ------------|*|-1 + ------------|
/ 3 \ / 3 \ \2/ \ \ 2 // | 2/a\ | | 2/a pi\|
cos|2*a + ----| cos|----| | cos |-| | | cos |- - --||
\ 2*pi/ \2*pi/ \ \2/ / \ \2 4 //
--------------- - --------- + -----------------------------------------------------------------
2 2 2
$$\frac{\left(-1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \left(-1 + \frac{\cos^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}\right) \left(\cos{\left(a - \frac{\pi}{2} \right)} + 1\right) \cos^{2}{\left(\frac{a}{2} \right)}}{2} + \frac{\cos{\left(2 a + \frac{3}{2 \pi} \right)}}{2} - \frac{\cos{\left(\frac{3}{2 \pi} \right)}}{2}$$
2/a\ /a 3 \
4*sin |-|*tan|- + ----|
/ 2/a\\ \2/ \2 4*pi/
|-1 + 2*cos |-||*sin(a) - ---------------------------
\ \2// / 2/a 3 \\ /a\
|1 + tan |- + ----||*tan|-|
\ \2 4*pi// \2/
$$\left(2 \cos^{2}{\left(\frac{a}{2} \right)} - 1\right) \sin{\left(a \right)} - \frac{4 \sin^{2}{\left(\frac{a}{2} \right)} \tan{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{3}{4 \pi} \right)} + 1\right) \tan{\left(\frac{a}{2} \right)}}$$
/ / 3 \ \ |- sin|a + ----| + cos(a)|*sin(a) \ \ 2*pi/ /
$$\left(- \sin{\left(a + \frac{3}{2 \pi} \right)} + \cos{\left(a \right)}\right) \sin{\left(a \right)}$$
1 1
------------- + ------------------------------
csc(a)*sec(a) / pi\ / pi 3 \
sec|a + --|*sec|a - -- + ----|
\ 2 / \ 2 2*pi/
$$\frac{1}{\sec{\left(a + \frac{\pi}{2} \right)} \sec{\left(a - \frac{\pi}{2} + \frac{3}{2 \pi} \right)}} + \frac{1}{\csc{\left(a \right)} \sec{\left(a \right)}}$$
1/(csc(a)*sec(a)) + 1/(sec(a + pi/2)*sec(a - pi/2 + 3/(2*pi)))