Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
(sin(a-pi)^2-4*cos(3*pi/2-a/2)^2)/(cos(a-5*pi/2)^2+4*cos(pi/2+a/2)^2-4)
¿Cómo vas a descomponer esta (sin(a-pi)^2-4*cos(3*pi/2-a/2)^2)/(cos(a-5*pi/2)^2+4*cos(pi/2+a/2)^2-4) expresión en fracciones?
Solución
Ha introducido
[src]
2 2/3*pi a\
sin (a - pi) - 4*cos |---- - -|
\ 2 2/
-----------------------------------
2/ 5*pi\ 2/pi a\
cos |a - ----| + 4*cos |-- + -| - 4
\ 2 / \2 2/
$$\frac{\sin^{2}{\left(a - \pi \right)} - 4 \cos^{2}{\left(- \frac{a}{2} + \frac{3 \pi}{2} \right)}}{\left(\cos^{2}{\left(a - \frac{5 \pi}{2} \right)} + 4 \cos^{2}{\left(\frac{a}{2} + \frac{\pi}{2} \right)}\right) - 4}$$
(sin(a - pi)^2 - 4*cos((3*pi)/2 - a/2)^2)/(cos(a - 5*pi/2)^2 + 4*cos(pi/2 + a/2)^2 - 4)
Simplificación general
[src]
3 - 4*cos(a) + cos(2*a) ----------------------- 3 + 4*cos(a) + cos(2*a)
$$\frac{- 4 \cos{\left(a \right)} + \cos{\left(2 a \right)} + 3}{4 \cos{\left(a \right)} + \cos{\left(2 a \right)} + 3}$$
(3 - 4*cos(a) + cos(2*a))/(3 + 4*cos(a) + cos(2*a))
Respuesta numérica
[src]
(sin(a - pi)^2 - 4.0*cos((3*pi)/2 - a/2)^2)/(-4.0 + cos(a - 5*pi/2)^2 + 4.0*cos(pi/2 + a/2)^2)
(sin(a - pi)^2 - 4.0*cos((3*pi)/2 - a/2)^2)/(-4.0 + cos(a - 5*pi/2)^2 + 4.0*cos(pi/2 + a/2)^2)
Denominador racional
[src]
2 2/a\
sin (a) - 4*sin |-|
\2/
------------------------------------
2/ 5*pi\ 2/pi a\
-4 + cos |a - ----| + 4*cos |-- + -|
\ 2 / \2 2/
$$\frac{- 4 \sin^{2}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{5 \pi}{2} \right)} + 4 \cos^{2}{\left(\frac{a}{2} + \frac{\pi}{2} \right)} - 4}$$
(sin(a)^2 - 4*sin(a/2)^2)/(-4 + cos(a - 5*pi/2)^2 + 4*cos(pi/2 + a/2)^2)
Potencias
[src]
2
/ /a 3*pi\ / a 3*pi\\
| I*|- - ----| I*|- - + ----|| 2
| \2 2 / \ 2 2 /| / I*(pi - a) I*(a - pi)\
|e e | \- e + e /
- 4*|------------- + ---------------| - ------------------------------
\ 2 2 / 4
-------------------------------------------------------------------------
2 2
/ / 5*pi\ / 5*pi\\ / /pi a\ / pi a\\
| I*|a - ----| I*|-a + ----|| | I*|-- + -| I*|- -- - -||
| \ 2 / \ 2 /| | \2 2/ \ 2 2/|
|e e | |e e |
-4 + |------------- + --------------| + 4*|----------- + -------------|
\ 2 2 / \ 2 2 /
$$\frac{- \frac{\left(- e^{i \left(\pi - a\right)} + e^{i \left(a - \pi\right)}\right)^{2}}{4} - 4 \left(\frac{e^{i \left(- \frac{a}{2} + \frac{3 \pi}{2}\right)}}{2} + \frac{e^{i \left(\frac{a}{2} - \frac{3 \pi}{2}\right)}}{2}\right)^{2}}{\left(\frac{e^{i \left(- a + \frac{5 \pi}{2}\right)}}{2} + \frac{e^{i \left(a - \frac{5 \pi}{2}\right)}}{2}\right)^{2} + 4 \left(\frac{e^{i \left(- \frac{a}{2} - \frac{\pi}{2}\right)}}{2} + \frac{e^{i \left(\frac{a}{2} + \frac{\pi}{2}\right)}}{2}\right)^{2} - 4}$$
2 2/a\
sin (a) - 4*sin |-|
\2/
------------------------
2 2/a\
-4 + sin (a) + 4*sin |-|
\2/
$$\frac{- 4 \sin^{2}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}}{4 \sin^{2}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)} - 4}$$
(sin(a)^2 - 4*sin(a/2)^2)/(-4 + sin(a)^2 + 4*sin(a/2)^2)
Combinatoria
[src]
/ /a\ \ / /a\ \
|- 2*sin|-| + sin(a)|*|2*sin|-| + sin(a)|
\ \2/ / \ \2/ /
-----------------------------------------
2 2/a\
-4 + sin (a) + 4*sin |-|
\2/
$$\frac{\left(- 2 \sin{\left(\frac{a}{2} \right)} + \sin{\left(a \right)}\right) \left(2 \sin{\left(\frac{a}{2} \right)} + \sin{\left(a \right)}\right)}{4 \sin^{2}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)} - 4}$$
(-2*sin(a/2) + sin(a))*(2*sin(a/2) + sin(a))/(-4 + sin(a)^2 + 4*sin(a/2)^2)
Unión de expresiones racionales
[src]
2 2/-a + 3*pi\
sin (a) - 4*cos |---------|
\ 2 /
---------------------------------------
2/-5*pi + 2*a\ 2/pi + a\
-4 + cos |-----------| + 4*cos |------|
\ 2 / \ 2 /
$$\frac{\sin^{2}{\left(a \right)} - 4 \cos^{2}{\left(\frac{- a + 3 \pi}{2} \right)}}{4 \cos^{2}{\left(\frac{a + \pi}{2} \right)} + \cos^{2}{\left(\frac{2 a - 5 \pi}{2} \right)} - 4}$$
(sin(a)^2 - 4*cos((-a + 3*pi)/2)^2)/(-4 + cos((-5*pi + 2*a)/2)^2 + 4*cos((pi + a)/2)^2)
Compilar la expresión
[src]
2 2/3*pi a\
sin (a - pi) - 4*cos |---- - -|
\ 2 2/
------------------------------------
2/ 5*pi\ 2/pi a\
-4 + cos |a - ----| + 4*cos |-- + -|
\ 2 / \2 2/
$$\frac{\sin^{2}{\left(a - \pi \right)} - 4 \cos^{2}{\left(- \frac{a}{2} + \frac{3 \pi}{2} \right)}}{\cos^{2}{\left(a - \frac{5 \pi}{2} \right)} + 4 \cos^{2}{\left(\frac{a}{2} + \frac{\pi}{2} \right)} - 4}$$
(sin(a - pi)^2 - 4*cos((3*pi)/2 - a/2)^2)/(-4 + cos(a - 5*pi/2)^2 + 4*cos(pi/2 + a/2)^2)
Denominador común
[src]
2 2/a\
sin (a - pi) - 4*sin |-|
\2/
------------------------
2 2/a\
-4 + sin (a) + 4*sin |-|
\2/
$$\frac{- 4 \sin^{2}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a - \pi \right)}}{4 \sin^{2}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)} - 4}$$
(sin(a - pi)^2 - 4*sin(a/2)^2)/(-4 + sin(a)^2 + 4*sin(a/2)^2)
Parte trigonométrica
[src]
1 4
-------------- - -------
2/ 3*pi\ 2/a\
sec |a - ----| csc |-|
\ 2 / \2/
------------------------
1 4
-4 + ------- + -------
2 2/a\
csc (a) csc |-|
\2/
$$\frac{\frac{1}{\sec^{2}{\left(a - \frac{3 \pi}{2} \right)}} - \frac{4}{\csc^{2}{\left(\frac{a}{2} \right)}}}{-4 + \frac{1}{\csc^{2}{\left(a \right)}} + \frac{4}{\csc^{2}{\left(\frac{a}{2} \right)}}}$$
2/ pi\ 2/ pi a\ 4/ pi a\
cos |a - --| - 16*cos |- -- + -| + 16*cos |- -- + -|
\ 2 / \ 2 4/ \ 2 4/
---------------------------------------------------------------------
2
/ 2/a pi\\
| cos |- + --|| 2
| \2 4 /| / / pi\\
|1 - ------------| *|1 + cos|a - --||
| 2/a pi\| \ \ 2 //
| cos |- - --||
4/a\ 2/a\ \ \2 4 //
-4 - 16*cos |-| + 16*cos |-| + --------------------------------------
\4/ \4/ 4
$$\frac{16 \cos^{4}{\left(\frac{a}{4} - \frac{\pi}{2} \right)} - 16 \cos^{2}{\left(\frac{a}{4} - \frac{\pi}{2} \right)} + \cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\frac{\left(1 - \frac{\cos^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}\right)^{2} \left(\cos{\left(a - \frac{\pi}{2} \right)} + 1\right)^{2}}{4} - 16 \cos^{4}{\left(\frac{a}{4} \right)} + 16 \cos^{2}{\left(\frac{a}{4} \right)} - 4}$$
/ / 2/a\\ 2/a\ \
| 2*|-1 + cot |-|| 4*cot |-| |
| \ \2// \2/ |
-|-2 + ---------------- + --------------|
| 2/a\ 2|
| 1 + cot |-| / 2/a\\ |
| \2/ |1 + cot |-|| |
\ \ \2// /
------------------------------------------
2
/ 2/a\\
| -1 + cot |-||
| \2/|
|1 + ------------|
| 2/a\ |
| 1 + cot |-| |
\ \2/ /
$$- \frac{\frac{2 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} - 2 + \frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}}{\left(\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} + 1\right)^{2}}$$
1 - cos(2*a) / /a\\ 4/a\ 8/a\
------------ - 8*|1 - cos|-|| + 256*cot |-|*sin |-|
2 \ \2// \8/ \8/
------------------------------------------------------------------------------------
2
/ 2/a pi\\ 2
4 |1 - cot |- + --|| *(1 + sin(a))
/ /a\\ / 2/a\\ 8/a\ \ \2 4 //
-4 + 8*|1 + cos|-|| - 16*|-1 + cot |-|| *sin |-| + ---------------------------------
\ \2// \ \8// \8/ 4
$$\frac{- 8 \left(1 - \cos{\left(\frac{a}{2} \right)}\right) + \frac{1 - \cos{\left(2 a \right)}}{2} + 256 \sin^{8}{\left(\frac{a}{8} \right)} \cot^{4}{\left(\frac{a}{8} \right)}}{\frac{\left(1 - \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}\right)^{2} \left(\sin{\left(a \right)} + 1\right)^{2}}{4} + 8 \left(\cos{\left(\frac{a}{2} \right)} + 1\right) - 16 \left(\cot^{2}{\left(\frac{a}{8} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{a}{8} \right)} - 4}$$
1 4
3 + -------- - ------
sec(2*a) sec(a)
---------------------
1 4
3 + -------- + ------
sec(2*a) sec(a)
$$\frac{3 + \frac{1}{\sec{\left(2 a \right)}} - \frac{4}{\sec{\left(a \right)}}}{3 + \frac{1}{\sec{\left(2 a \right)}} + \frac{4}{\sec{\left(a \right)}}}$$
/ pi\ /pi \
3 - 4*sin|a + --| + sin|-- + 2*a|
\ 2 / \2 /
---------------------------------
/ pi\ /pi \
3 + 4*sin|a + --| + sin|-- + 2*a|
\ 2 / \2 /
$$\frac{- 4 \sin{\left(a + \frac{\pi}{2} \right)} + \sin{\left(2 a + \frac{\pi}{2} \right)} + 3}{4 \sin{\left(a + \frac{\pi}{2} \right)} + \sin{\left(2 a + \frac{\pi}{2} \right)} + 3}$$
/ / 2/a\\ 2/a\ \
| 2*|1 - tan |-|| 4*tan |-| |
| \ \2// \2/ |
-|-2 + --------------- + --------------|
| 2/a\ 2|
| 1 + tan |-| / 2/a\\ |
| \2/ |1 + tan |-|| |
\ \ \2// /
-----------------------------------------
2
/ 2/a\\
| 1 - tan |-||
| \2/|
|1 + -----------|
| 2/a\|
| 1 + tan |-||
\ \2//
$$- \frac{\frac{2 \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} - 2 + \frac{4 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}}{\left(\frac{1 - \tan^{2}{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + 1\right)^{2}}$$
2
/ 2/pi a\\ 2/a\
4*|1 - tan |-- + -|| 4*cot |-|
\ \4 4// \2/
- --------------------- + --------------
2 2
/ 2/pi a\\ / 2/a\\
|1 + tan |-- + -|| |1 + cot |-||
\ \4 4// \ \2//
------------------------------------------------
2 2
/ 2/a pi\\ / 2/pi a\\
|1 - cot |- + --|| 4*|1 - tan |-- + -||
\ \2 4 // \ \4 4//
-4 + ------------------- + ---------------------
2 2
/ 2/a pi\\ / 2/pi a\\
|1 + cot |- + --|| |1 + tan |-- + -||
\ \2 4 // \ \4 4//
$$\frac{- \frac{4 \left(1 - \tan^{2}{\left(\frac{a}{4} + \frac{\pi}{4} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{4} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}}{\frac{4 \left(1 - \tan^{2}{\left(\frac{a}{4} + \frac{\pi}{4} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{4} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{\left(1 - \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} - 4}$$
2/a\ 2/a\ 4/a\
64*tan |-| 4*tan |-| 256*tan |-|
\8/ \2/ \8/
- -------------- + -------------- + --------------
2 2 4
/ 2/a\\ / 2/a\\ / 2/a\\
|1 + tan |-|| |1 + tan |-|| |1 + tan |-||
\ \8// \ \2// \ \8//
-----------------------------------------------------------------------------------
2
/ /a\ \
2 | 2*tan|-| |
/ 1 \ | \2/ |
4 2 |1 - ------------| *|1 + -----------|
/ 2/a\\ / 2/a\\ | 2/a pi\| | 2/a\|
16*|1 - tan |-|| 16*|1 - tan |-|| | tan |- + --|| | 1 + tan |-||
\ \8// \ \8// \ \2 4 // \ \2//
-4 - ----------------- + ----------------- + --------------------------------------
4 2 4
/ 2/a\\ / 2/a\\
|1 + tan |-|| |1 + tan |-||
\ \8// \ \8//
$$\frac{\frac{4 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} - \frac{64 \tan^{2}{\left(\frac{a}{8} \right)}}{\left(\tan^{2}{\left(\frac{a}{8} \right)} + 1\right)^{2}} + \frac{256 \tan^{4}{\left(\frac{a}{8} \right)}}{\left(\tan^{2}{\left(\frac{a}{8} \right)} + 1\right)^{4}}}{\frac{\left(1 + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right)^{2} \left(1 - \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}\right)^{2}}{4} - \frac{16 \left(1 - \tan^{2}{\left(\frac{a}{8} \right)}\right)^{4}}{\left(\tan^{2}{\left(\frac{a}{8} \right)} + 1\right)^{4}} + \frac{16 \left(1 - \tan^{2}{\left(\frac{a}{8} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{8} \right)} + 1\right)^{2}} - 4}$$
2 2/a\
sin (a) - 4*sin |-|
\2/
------------------------
2 2/a\
-4 + sin (a) + 4*sin |-|
\2/
$$\frac{- 4 \sin^{2}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}}{4 \sin^{2}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)} - 4}$$
2
/ 2/pi a\\
4*|1 - tan |-- + -||
\ \4 4// 4
- --------------------- + ----------------------
2 2
/ 2/pi a\\ / 1 \ 2/a\
|1 + tan |-- + -|| |1 + -------| *tan |-|
\ \4 4// | 2/a\| \2/
| tan |-||
\ \2//
------------------------------------------------
2
/ 1 \
|1 - ------------| 2
| 2/a pi\| / 2/pi a\\
| tan |- + --|| 4*|1 - tan |-- + -||
\ \2 4 // \ \4 4//
-4 + ------------------- + ---------------------
2 2
/ 1 \ / 2/pi a\\
|1 + ------------| |1 + tan |-- + -||
| 2/a pi\| \ \4 4//
| tan |- + --||
\ \2 4 //
$$\frac{- \frac{4 \left(1 - \tan^{2}{\left(\frac{a}{4} + \frac{\pi}{4} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{4} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{4}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \tan^{2}{\left(\frac{a}{2} \right)}}}{\frac{4 \left(1 - \tan^{2}{\left(\frac{a}{4} + \frac{\pi}{4} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{4} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{\left(1 - \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}\right)^{2}} - 4}$$
2 2/a pi\
sin (a) - 4*cos |- - --|
\2 2 /
----------------------------------
2/ pi\ 2/a pi\
-4 + cos |a - --| + 4*cos |- - --|
\ 2 / \2 2 /
$$\frac{\sin^{2}{\left(a \right)} - 4 \cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{4 \cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)} + \cos^{2}{\left(a - \frac{\pi}{2} \right)} - 4}$$
3 - 4*cos(a) + cos(2*a) ----------------------- 3 + 4*cos(a) + cos(2*a)
$$\frac{- 4 \cos{\left(a \right)} + \cos{\left(2 a \right)} + 3}{4 \cos{\left(a \right)} + \cos{\left(2 a \right)} + 3}$$
1 4
3 + ------------- - -----------
/pi \ /pi \
csc|-- - 2*a| csc|-- - a|
\2 / \2 /
-------------------------------
1 4
3 + ------------- + -----------
/pi \ /pi \
csc|-- - 2*a| csc|-- - a|
\2 / \2 /
$$\frac{3 - \frac{4}{\csc{\left(- a + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}}}{3 + \frac{4}{\csc{\left(- a + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}}}$$
2/a\ 2/a\ 4/a\
64*cot |-| 4*cot |-| 256*cot |-|
\8/ \2/ \8/
- -------------- + -------------- + --------------
2 2 4
/ 2/a\\ / 2/a\\ / 2/a\\
|1 + cot |-|| |1 + cot |-|| |1 + cot |-||
\ \8// \ \2// \ \8//
-------------------------------------------------------------------------------------
2
/ /a\ \
2 | 2*cot|-| |
/ 2/a pi\\ | \2/ |
4 2 |1 - cot |- + --|| *|1 + -----------|
/ 2/a\\ / 2/a\\ \ \2 4 // | 2/a\|
16*|-1 + cot |-|| 16*|-1 + cot |-|| | 1 + cot |-||
\ \8// \ \8// \ \2//
-4 - ------------------ + ------------------ + --------------------------------------
4 2 4
/ 2/a\\ / 2/a\\
|1 + cot |-|| |1 + cot |-||
\ \8// \ \8//
$$\frac{\frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} - \frac{64 \cot^{2}{\left(\frac{a}{8} \right)}}{\left(\cot^{2}{\left(\frac{a}{8} \right)} + 1\right)^{2}} + \frac{256 \cot^{4}{\left(\frac{a}{8} \right)}}{\left(\cot^{2}{\left(\frac{a}{8} \right)} + 1\right)^{4}}}{\frac{\left(1 + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right)^{2} \left(1 - \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}\right)^{2}}{4} - \frac{16 \left(\cot^{2}{\left(\frac{a}{8} \right)} - 1\right)^{4}}{\left(\cot^{2}{\left(\frac{a}{8} \right)} + 1\right)^{4}} + \frac{16 \left(\cot^{2}{\left(\frac{a}{8} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{8} \right)} + 1\right)^{2}} - 4}$$
2 2/a\ 4/a\
sin (a) - 16*sin |-| + 16*sin |-|
\4/ \4/
---------------------------------
2 2/a\
-4 + sin (a) + 4*sin |-|
\2/
$$\frac{16 \sin^{4}{\left(\frac{a}{4} \right)} - 16 \sin^{2}{\left(\frac{a}{4} \right)} + \sin^{2}{\left(a \right)}}{4 \sin^{2}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)} - 4}$$
1 - cos(2*a) / /a\\ 8/a\ 4/a\
------------ - 8*|1 - cos|-|| + 256*cos |-|*tan |-|
2 \ \2// \8/ \8/
-----------------------------------------------------------------------------------
2
/ 1 \ 2
|1 - ------------| *(1 + sin(a))
| 2/a pi\|
4 | tan |- + --||
/ /a\\ / 2/a\\ 8/a\ \ \2 4 //
-4 + 8*|1 + cos|-|| - 16*|1 - tan |-|| *cos |-| + ---------------------------------
\ \2// \ \8// \8/ 4
$$\frac{- 8 \left(1 - \cos{\left(\frac{a}{2} \right)}\right) + \frac{1 - \cos{\left(2 a \right)}}{2} + 256 \cos^{8}{\left(\frac{a}{8} \right)} \tan^{4}{\left(\frac{a}{8} \right)}}{- 16 \left(1 - \tan^{2}{\left(\frac{a}{8} \right)}\right)^{4} \cos^{8}{\left(\frac{a}{8} \right)} + \frac{\left(1 - \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}\right)^{2} \left(\sin{\left(a \right)} + 1\right)^{2}}{4} + 8 \left(\cos{\left(\frac{a}{2} \right)} + 1\right) - 4}$$
2 2/a\ 4/a\
sin (a) - 16*sin |-| + 16*sin |-|
\4/ \4/
----------------------------------------------------------------
2
/ 2/a pi\\ 2
|1 - cot |- + --|| *(1 + sin(a))
4/a\ 2/a\ \ \2 4 //
-4 - 16*cos |-| + 16*cos |-| + ---------------------------------
\4/ \4/ 4
$$\frac{16 \sin^{4}{\left(\frac{a}{4} \right)} - 16 \sin^{2}{\left(\frac{a}{4} \right)} + \sin^{2}{\left(a \right)}}{\frac{\left(1 - \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}\right)^{2} \left(\sin{\left(a \right)} + 1\right)^{2}}{4} - 16 \cos^{4}{\left(\frac{a}{4} \right)} + 16 \cos^{2}{\left(\frac{a}{4} \right)} - 4}$$
1 4
------- - ------------
2 2/pi a\
csc (a) sec |-- - -|
\2 2/
--------------------------------
1 4
-4 + ------------ + ------------
2/pi \ 2/pi a\
sec |-- - a| sec |-- - -|
\2 / \2 2/
$$\frac{- \frac{4}{\sec^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{2}{\left(a \right)}}}{-4 + \frac{4}{\sec^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + \frac{1}{\sec^{2}{\left(- a + \frac{\pi}{2} \right)}}}$$
2
/ 2/pi a\\
4*|-1 + cot |-- + -||
\ \4 4// 4
- ---------------------- + ----------------------
2 2
/ 2/pi a\\ / 1 \ 2/a\
|1 + cot |-- + -|| |1 + -------| *cot |-|
\ \4 4// | 2/a\| \2/
| cot |-||
\ \2//
--------------------------------------------------
2
/ 1 \
|-1 + ------------| 2
| 2/a pi\| / 2/pi a\\
| cot |- + --|| 4*|-1 + cot |-- + -||
\ \2 4 // \ \4 4//
-4 + -------------------- + ----------------------
2 2
/ 1 \ / 2/pi a\\
|1 + ------------| |1 + cot |-- + -||
| 2/a pi\| \ \4 4//
| cot |- + --||
\ \2 4 //
$$\frac{- \frac{4 \left(\cot^{2}{\left(\frac{a}{4} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{4} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{4}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \cot^{2}{\left(\frac{a}{2} \right)}}}{\frac{\left(-1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}\right)^{2}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}\right)^{2}} + \frac{4 \left(\cot^{2}{\left(\frac{a}{4} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{4} + \frac{\pi}{4} \right)} + 1\right)^{2}} - 4}$$
2 2/a\ 4/a\
sin (a) - 16*sin |-| + 16*sin |-|
\4/ \4/
----------------------------------------------------------------------------
2
/ 2 \
| cos (a) | 2
|1 - --------------| *(1 + sin(a))
| 4/a pi\|
| 4*sin |- + --||
4/pi a\ 2/pi a\ \ \2 4 //
-4 - 16*sin |-- + -| + 16*sin |-- + -| + -----------------------------------
\2 4/ \2 4/ 4
$$\frac{16 \sin^{4}{\left(\frac{a}{4} \right)} - 16 \sin^{2}{\left(\frac{a}{4} \right)} + \sin^{2}{\left(a \right)}}{\frac{\left(1 - \frac{\cos^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}\right)^{2} \left(\sin{\left(a \right)} + 1\right)^{2}}{4} - 16 \sin^{4}{\left(\frac{a}{4} + \frac{\pi}{2} \right)} + 16 \sin^{2}{\left(\frac{a}{4} + \frac{\pi}{2} \right)} - 4}$$
/ 2/a\\
2 4*|-1 + cot |-||
-1 + cot (a) \ \2//
3 + ------------ - ----------------
2 2/a\
1 + cot (a) 1 + cot |-|
\2/
-----------------------------------
/ 2/a\\
2 4*|-1 + cot |-||
-1 + cot (a) \ \2//
3 + ------------ + ----------------
2 2/a\
1 + cot (a) 1 + cot |-|
\2/
$$\frac{- \frac{4 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} + \frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} + 3}{\frac{4 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} + \frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} + 3}$$
2 2/ a\
sin (a) - 4*sin |2*pi - -|
\ 2/
------------------------------------
2 2/ a\
-4 + sin (a - 2*pi) + 4*sin |pi + -|
\ 2/
$$\frac{\sin^{2}{\left(a \right)} - 4 \sin^{2}{\left(- \frac{a}{2} + 2 \pi \right)}}{4 \sin^{2}{\left(\frac{a}{2} + \pi \right)} + \sin^{2}{\left(a - 2 \pi \right)} - 4}$$
/ 2 \
-\-2 + sin (a) + 2*cos(a)/
---------------------------
2
(1 + cos(a))
$$- \frac{\sin^{2}{\left(a \right)} + 2 \cos{\left(a \right)} - 2}{\left(\cos{\left(a \right)} + 1\right)^{2}}$$
1 16 16
------------ - -------------- + --------------
2/ pi\ 2/ pi a\ 4/ pi a\
sec |a - --| sec |- -- + -| sec |- -- + -|
\ 2 / \ 2 4/ \ 2 4/
---------------------------------------------------------------
2
/ 2/a pi\\
2 | sec |- - --||
/ 1 \ | \2 4 /|
|1 + -----------| *|1 - ------------|
| / pi\| | 2/a pi\|
| sec|a - --|| | sec |- + --||
16 16 \ \ 2 // \ \2 4 //
-4 - ------- + ------- + --------------------------------------
4/a\ 2/a\ 4
sec |-| sec |-|
\4/ \4/
$$\frac{\frac{1}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} - \frac{16}{\sec^{2}{\left(\frac{a}{4} - \frac{\pi}{2} \right)}} + \frac{16}{\sec^{4}{\left(\frac{a}{4} - \frac{\pi}{2} \right)}}}{\frac{\left(1 + \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}\right)^{2} \left(- \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}{\sec^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} + 1\right)^{2}}{4} - 4 + \frac{16}{\sec^{2}{\left(\frac{a}{4} \right)}} - \frac{16}{\sec^{4}{\left(\frac{a}{4} \right)}}}$$
/ 2/ pi\ \
-|-2 + cos |a - --| + 2*cos(a)|
\ \ 2 / /
--------------------------------
2
(1 + cos(a))
$$- \frac{2 \cos{\left(a \right)} + \cos^{2}{\left(a - \frac{\pi}{2} \right)} - 2}{\left(\cos{\left(a \right)} + 1\right)^{2}}$$
/ 2/a\\
2 4*|1 - tan |-||
1 - tan (a) \ \2//
3 + ----------- - ---------------
2 2/a\
1 + tan (a) 1 + tan |-|
\2/
---------------------------------
/ 2/a\\
2 4*|1 - tan |-||
1 - tan (a) \ \2//
3 + ----------- + ---------------
2 2/a\
1 + tan (a) 1 + tan |-|
\2/
$$\frac{- \frac{4 \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + \frac{1 - \tan^{2}{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} + 3}{\frac{4 \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + \frac{1 - \tan^{2}{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} + 3}$$
2
/ 2/pi a\\ 2/a\
4*|-1 + cot |-- + -|| 4*tan |-|
\ \4 4// \2/
- ---------------------- + --------------
2 2
/ 2/pi a\\ / 2/a\\
|1 + cot |-- + -|| |1 + tan |-||
\ \4 4// \ \2//
--------------------------------------------------
2 2
/ 2/a pi\\ / 2/pi a\\
|-1 + tan |- + --|| 4*|-1 + cot |-- + -||
\ \2 4 // \ \4 4//
-4 + -------------------- + ----------------------
2 2
/ 2/a pi\\ / 2/pi a\\
|1 + tan |- + --|| |1 + cot |-- + -||
\ \2 4 // \ \4 4//
$$\frac{- \frac{4 \left(\cot^{2}{\left(\frac{a}{4} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{4} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{4 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}}{\frac{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{4 \left(\cot^{2}{\left(\frac{a}{4} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{4} + \frac{\pi}{4} \right)} + 1\right)^{2}} - 4}$$
2/ 3*pi\ 2/a\
cos |a - ----| - 4*sin |-|
\ 2 / \2/
--------------------------
2 2/a\
-4 + sin (a) + 4*sin |-|
\2/
$$\frac{- 4 \sin^{2}{\left(\frac{a}{2} \right)} + \cos^{2}{\left(a - \frac{3 \pi}{2} \right)}}{4 \sin^{2}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)} - 4}$$
2
/a\ / /a\\
15 - 16*cos|-| - 8*|1 - cos|-|| + cos(2*a)
\2/ \ \2//
-------------------------------------------
3 + 4*cos(a) + cos(2*a)
$$\frac{- 8 \left(1 - \cos{\left(\frac{a}{2} \right)}\right)^{2} - 16 \cos{\left(\frac{a}{2} \right)} + \cos{\left(2 a \right)} + 15}{4 \cos{\left(a \right)} + \cos{\left(2 a \right)} + 3}$$
/ 1 2 \
-|-2 + ------------ + ------|
| 2/ pi\ sec(a)|
| sec |a - --| |
\ \ 2 / /
------------------------------
2
/ 1 \
|1 + ------|
\ sec(a)/
$$- \frac{-2 + \frac{1}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + \frac{2}{\sec{\left(a \right)}}}{\left(1 + \frac{1}{\sec{\left(a \right)}}\right)^{2}}$$
1 16 16
------- - ------- + -------
2 2/a\ 4/a\
csc (a) csc |-| csc |-|
\4/ \4/
----------------------------------------------------------------------
2
/ 2/a pi\ \
2 | csc |- + --| |
/ 1 \ | \2 4 / |
|1 + ------| *|1 - --------------|
\ csc(a)/ | 2/ a pi\|
| csc |- - + --||
16 16 \ \ 2 4 //
-4 - ------------ + ------------ + -----------------------------------
4/pi a\ 2/pi a\ 4
csc |-- - -| csc |-- - -|
\2 4/ \2 4/
$$\frac{\frac{1}{\csc^{2}{\left(a \right)}} - \frac{16}{\csc^{2}{\left(\frac{a}{4} \right)}} + \frac{16}{\csc^{4}{\left(\frac{a}{4} \right)}}}{\frac{\left(1 - \frac{\csc^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{4} \right)}}\right)^{2} \left(1 + \frac{1}{\csc{\left(a \right)}}\right)^{2}}{4} - 4 + \frac{16}{\csc^{2}{\left(- \frac{a}{4} + \frac{\pi}{2} \right)}} - \frac{16}{\csc^{4}{\left(- \frac{a}{4} + \frac{\pi}{2} \right)}}}$$
/ 2 / pi\\
-|-2 + sin (a) + 2*sin|a + --||
\ \ 2 //
--------------------------------
2
/ / pi\\
|1 + sin|a + --||
\ \ 2 //
$$- \frac{\sin^{2}{\left(a \right)} + 2 \sin{\left(a + \frac{\pi}{2} \right)} - 2}{\left(\sin{\left(a + \frac{\pi}{2} \right)} + 1\right)^{2}}$$
/ 1 2 \
-|-2 + ------- + -----------|
| 2 /pi \|
| csc (a) csc|-- - a||
\ \2 //
------------------------------
2
/ 1 \
|1 + -----------|
| /pi \|
| csc|-- - a||
\ \2 //
$$- \frac{-2 + \frac{2}{\csc{\left(- a + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{2}{\left(a \right)}}}{\left(1 + \frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}}\right)^{2}}$$
1 4
------- - -------
2 2/a\
csc (a) csc |-|
\2/
----------------------
1 4
-4 + ------- + -------
2 2/a\
csc (a) csc |-|
\2/
$$\frac{\frac{1}{\csc^{2}{\left(a \right)}} - \frac{4}{\csc^{2}{\left(\frac{a}{2} \right)}}}{-4 + \frac{1}{\csc^{2}{\left(a \right)}} + \frac{4}{\csc^{2}{\left(\frac{a}{2} \right)}}}$$
(csc(a)^(-2) - 4/csc(a/2)^2)/(-4 + csc(a)^(-2) + 4/csc(a/2)^2)