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