Господин Экзамен
Другие калькуляторы
-
Как пользоваться?
- sin(4*a)+cos(2*a)+sin(2*a)*cos(2*a) если a=-2
- 1-cos(t)^2/1-sin(t)^2 если t=-4
- (4*x^2-40*x*y+100*y^2)/(15*y-3*x) если y=2
- a+3 если a=-2
- Разложить многочлен на множители:
- 49*k^2+42*k*p+9*p^2
- 4*c^2*d^2+36*c^2*d^3+6*c*d^7
- 25-10*x+x^2
- a^3+b^6
- Общий знаменатель:
- (a-b)*2+a*b/(a+b)*2-a*b/a5+a3*b2+a2*b3+b5/(a3+a2*b+b2*a+b3)*(a3-b3)
- (((d-9)/(d))-((c-9)/(c)))*((c*d)/(d^2-c^2))
- 2*cos(-2*pi-b)+sin(-pi/2+b)/2*cos(b+pi)
- pi/2-pi/4
- Умножение столбиком:
- 773*797
- 36*25
- 90*15
- 70*13
- Деление столбиком:
- 11/4
- 65376/2
- 876/7
- 12240/3
- Раскрыть скобки в:
- (y+7)^2-2*(y+10)*(y+4)
- 5^2*p*q^2*(-4)^2*q*p
- (a+1)*(a-1)
- (2*k-3)*(3+2*k)-(3*k-5)^2-30*k
- Разложение числа на простые множители:
- 388
- 4860
- 6652
- 23716
- График функции y =:
- -2
- Производная:
- -2
- Интеграл d{x}:
-
-2
-
Идентичные выражения
- sin(четыре *a)+cos(два *a)+sin(два *a)*cos(два *a) если a=- два
- синус от (4 умножить на a) плюс косинус от (2 умножить на a) плюс синус от (2 умножить на a) умножить на косинус от (2 умножить на a) если a равно минус 2
- синус от (четыре умножить на a) плюс косинус от (два умножить на a) плюс синус от (два умножить на a) умножить на косинус от (два умножить на a) если a равно минус два
- sin(4a)+cos(2a)+sin(2a)cos(2a) если a=-2
- sin4a+cos2a+sin2acos2a если a=-2
- sin(4*a)+cos(2*a)+sin(2*a)*cos(2*a) при a=-2
-
Похожие выражения
- sin(4*a)+cos(2*a)-sin(2*a)*cos(2*a) если a=-2
- sin(4*a)-cos(2*a)+sin(2*a)*cos(2*a) если a=-2
- sin(4*a)+cos(2*a)+sin(2*a)*cos(2*a) если a=+2
sin(4*a)+cos(2*a)+sin(2*a)*cos(2*a) если a=-2
Решение
Вы ввели
[src]
sin(4*a) + cos(2*a) + sin(2*a)*cos(2*a)
$$\sin{\left(2 a \right)} \cos{\left(2 a \right)} + \sin{\left(4 a \right)} + \cos{\left(2 a \right)}$$
sin(4*a) + cos(2*a) + sin(2*a)*cos(2*a)
Общее упрощение
[src]
3*sin(4*a)
---------- + cos(2*a)
2
$$\frac{3 \sin{\left(4 a \right)}}{2} + \cos{\left(2 a \right)}$$
3*sin(4*a)/2 + cos(2*a)
Подстановка условия
[src]
sin(4*a) + cos(2*a) + sin(2*a)*cos(2*a) при a = -2
подставляем
sin(4*a) + cos(2*a) + sin(2*a)*cos(2*a)
$$\sin{\left(2 a \right)} \cos{\left(2 a \right)} + \sin{\left(4 a \right)} + \cos{\left(2 a \right)}$$
3*sin(4*a)
---------- + cos(2*a)
2
$$\frac{3 \sin{\left(4 a \right)}}{2} + \cos{\left(2 a \right)}$$
переменные
a = -2
$$a = -2$$
3*sin(4*(-2))
------------- + cos(2*(-2))
2
$$\frac{3 \sin{\left(4 (-2) \right)}}{2} + \cos{\left(2 (-2) \right)}$$
3*sin(8)
- -------- + cos(4)
2
$$- \frac{3 \sin{\left(8 \right)}}{2} + \cos{\left(4 \right)}$$
-3*sin(8)/2 + cos(4)
Степени
[src]
/ -2*I*a 2*I*a\
|e e | / -2*I*a 2*I*a\
-2*I*a 2*I*a / -4*I*a 4*I*a\ I*|------- + ------|*\- e + e /
e e I*\- e + e / \ 2 2 /
------- + ------ - ---------------------- - -----------------------------------------
2 2 2 2
$$- \frac{i \left(\frac{e^{2 i a}}{2} + \frac{e^{- 2 i a}}{2}\right) \left(e^{2 i a} - e^{- 2 i a}\right)}{2} + \frac{e^{2 i a}}{2} - \frac{i \left(e^{4 i a} - e^{- 4 i a}\right)}{2} + \frac{e^{- 2 i a}}{2}$$
exp(-2*i*a)/2 + exp(2*i*a)/2 - i*(-exp(-4*i*a) + exp(4*i*a))/2 - i*(exp(-2*i*a)/2 + exp(2*i*a)/2)*(-exp(-2*i*a) + exp(2*i*a))/2
Собрать выражение
[src]
3*sin(4*a)
---------- + cos(2*a)
2
$$\frac{3 \sin{\left(4 a \right)}}{2} + \cos{\left(2 a \right)}$$
3*sin(4*a)/2 + cos(2*a)
Тригонометрическая часть
[src]
3*sin(4*a)
---------- + cos(2*a)
2
$$\frac{3 \sin{\left(4 a \right)}}{2} + \cos{\left(2 a \right)}$$
/ pi\
3*cos|4*a - --|
\ 2 /
--------------- + cos(2*a)
2
$$\cos{\left(2 a \right)} + \frac{3 \cos{\left(4 a - \frac{\pi}{2} \right)}}{2}$$
3*sin(4*a) /pi \
---------- + sin|-- + 2*a|
2 \2 /
$$\frac{3 \sin{\left(4 a \right)}}{2} + \sin{\left(2 a + \frac{\pi}{2} \right)}$$
1 3
-------- + ---------------
sec(2*a) / pi\
2*sec|4*a - --|
\ 2 /
$$\frac{3}{2 \sec{\left(4 a - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(2 a \right)}}$$
1 3 ------------- + ---------- /pi \ 2*csc(4*a) csc|-- - 2*a| \2 /
$$\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} + \frac{3}{2 \csc{\left(4 a \right)}}$$
1 1 1 -------- + -------- + ----------------- csc(4*a) sec(2*a) csc(2*a)*sec(2*a)
$$\frac{1}{\sec{\left(2 a \right)}} + \frac{1}{\csc{\left(4 a \right)}} + \frac{1}{\csc{\left(2 a \right)} \sec{\left(2 a \right)}}$$
/pi \ /pi \
sin(2*a)*sin|-- + 2*a| + sin(4*a) + sin|-- + 2*a|
\2 / \2 /
$$\sin{\left(2 a \right)} \sin{\left(2 a + \frac{\pi}{2} \right)} + \sin{\left(4 a \right)} + \sin{\left(2 a + \frac{\pi}{2} \right)}$$
/ pi\ / pi\
cos(2*a)*cos|2*a - --| + cos(2*a) + cos|4*a - --|
\ 2 / \ 2 /
$$\cos{\left(2 a \right)} \cos{\left(2 a - \frac{\pi}{2} \right)} + \cos{\left(2 a \right)} + \cos{\left(4 a - \frac{\pi}{2} \right)}$$
2
1 - tan (a) 3*tan(2*a)
----------- + -------------
2 2
1 + tan (a) 1 + tan (2*a)
$$\frac{- \tan^{2}{\left(a \right)} + 1}{\tan^{2}{\left(a \right)} + 1} + \frac{3 \tan{\left(2 a \right)}}{\tan^{2}{\left(2 a \right)} + 1}$$
2 2 -1 - sin(2*a) + 2*cos (a) + 2*cos (a)*sin(2*a) + sin(4*a)
$$2 \sin{\left(2 a \right)} \cos^{2}{\left(a \right)} + 2 \cos^{2}{\left(a \right)} - \sin{\left(2 a \right)} + \sin{\left(4 a \right)} - 1$$
1 1 1
-------- + ------------- + ----------------------
sec(2*a) / pi\ / pi\
sec|4*a - --| sec(2*a)*sec|2*a - --|
\ 2 / \ 2 /
$$\frac{1}{\sec{\left(4 a - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(2 a \right)}} + \frac{1}{\sec{\left(2 a \right)} \sec{\left(2 a - \frac{\pi}{2} \right)}}$$
1 1 1
-------- + ------------- + ----------------------
csc(4*a) /pi \ /pi \
csc|-- - 2*a| csc(2*a)*csc|-- - 2*a|
\2 / \2 /
$$\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(4 a \right)}} + \frac{1}{\csc{\left(2 a \right)} \csc{\left(- 2 a + \frac{\pi}{2} \right)}}$$
1 1 1
-------- + ------------- + ----------------------
sec(2*a) /pi \ /pi \
sec|-- - 4*a| sec(2*a)*sec|-- - 2*a|
\2 / \2 /
$$\frac{1}{\sec{\left(- 4 a + \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(2 a \right)}} + \frac{1}{\sec{\left(2 a \right)} \sec{\left(- 2 a + \frac{\pi}{2} \right)}}$$
2 2 / 2 2 \ cos (a) - sin (a) + 6*\cos (a) - sin (a)/*cos(a)*sin(a)
$$6 \left(- \sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}\right) \sin{\left(a \right)} \cos{\left(a \right)} - \sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}$$
1 1 1
------------- + ------------- + ---------------------------
csc(pi - 4*a) /pi \ /pi \
csc|-- - 2*a| csc(pi - 2*a)*csc|-- - 2*a|
\2 / \2 /
$$\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(- 4 a + \pi \right)}} + \frac{1}{\csc{\left(- 2 a + \pi \right)} \csc{\left(- 2 a + \frac{\pi}{2} \right)}}$$
// 0 for 4*a mod pi = 0\
3*|< |
\\sin(4*a) otherwise / // 1 for a mod pi = 0\
--------------------------------- + |< |
2 \\cos(2*a) otherwise /
$$\left(\frac{3 \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right)}{2}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)$$
2 / 2 \
1 - tan (a) 2*tan(2*a) 2*\1 - tan (a)/*tan(a)
----------- + ------------- + ----------------------
2 2 2
1 + tan (a) 1 + tan (2*a) / 2 \
\1 + tan (a)/
$$\frac{- \tan^{2}{\left(a \right)} + 1}{\tan^{2}{\left(a \right)} + 1} + \frac{2 \cdot \left(- \tan^{2}{\left(a \right)} + 1\right) \tan{\left(a \right)}}{\left(\tan^{2}{\left(a \right)} + 1\right)^{2}} + \frac{2 \tan{\left(2 a \right)}}{\tan^{2}{\left(2 a \right)} + 1}$$
1 / 1 \
1 - ------- 2*|1 - -------|
2 | 2 |
cot (a) 2 \ cot (a)/
----------- + ------------------------ + ---------------------
1 / 1 \ 2
1 + ------- |1 + ---------|*cot(2*a) / 1 \
2 | 2 | |1 + -------| *cot(a)
cot (a) \ cot (2*a)/ | 2 |
\ cot (a)/
$$\frac{1 - \frac{1}{\cot^{2}{\left(a \right)}}}{1 + \frac{1}{\cot^{2}{\left(a \right)}}} + \frac{2 \cdot \left(1 - \frac{1}{\cot^{2}{\left(a \right)}}\right)}{\left(1 + \frac{1}{\cot^{2}{\left(a \right)}}\right)^{2} \cot{\left(a \right)}} + \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(2 a \right)}}\right) \cot{\left(2 a \right)}}$$
/ pi\ / pi\
2*tan|a + --| 4*tan(a)*tan|a + --|
2*tan(2*a) \ 4 / \ 4 /
------------- + ---------------- + --------------------------------
2 2/ pi\ / 2 \ / 2/ pi\\
1 + tan (2*a) 1 + tan |a + --| \1 + tan (a)/*|1 + tan |a + --||
\ 4 / \ \ 4 //
$$\frac{2 \tan{\left(a + \frac{\pi}{4} \right)}}{\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1} + \frac{2 \tan{\left(2 a \right)}}{\tan^{2}{\left(2 a \right)} + 1} + \frac{4 \tan{\left(a \right)} \tan{\left(a + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(a \right)} + 1\right) \left(\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1\right)}$$
/ pi\ / pi\
2*tan|a + --| 4*cot(a)*tan|a + --|
2*cot(2*a) \ 4 / \ 4 /
------------- + ---------------- + --------------------------------
2 2/ pi\ / 2 \ / 2/ pi\\
1 + cot (2*a) 1 + tan |a + --| \1 + cot (a)/*|1 + tan |a + --||
\ 4 / \ \ 4 //
$$\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} + \frac{2 \tan{\left(a + \frac{\pi}{4} \right)}}{\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1} + \frac{4 \tan{\left(a + \frac{\pi}{4} \right)} \cot{\left(a \right)}}{\left(\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(a \right)} + 1\right)}$$
// 0 for 4*a mod pi = 0\
|| |
|| 2*cot(2*a) |
3*|<------------- otherwise |
|| 2 | // 1 for a mod pi = 0\
||1 + cot (2*a) | || |
\\ / || 2 |
-------------------------------------- + |<-1 + cot (a) |
2 ||------------ otherwise |
|| 2 |
\\1 + cot (a) /
$$\left(\frac{3 \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right)}{2}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)$$
2/ pi\ / 2 \ / 2/ pi\\
2 -1 + tan |2*a + --| \-1 + cot (a)/*|-1 + tan |a + --||
-1 + cot (a) \ 4 / \ \ 4 //
------------ + ------------------- + ----------------------------------
2 2/ pi\ / 2 \ / 2/ pi\\
1 + cot (a) 1 + tan |2*a + --| \1 + cot (a)/*|1 + tan |a + --||
\ 4 / \ \ 4 //
$$\frac{\left(\tan^{2}{\left(a + \frac{\pi}{4} \right)} - 1\right) \left(\cot^{2}{\left(a \right)} - 1\right)}{\left(\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(a \right)} + 1\right)} + \frac{\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1} + \frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1}$$
2/ pi\ / 2/ pi\\ / 2 \
1 - cot |2*a + --| 2 |1 - cot |a + --||*\1 - tan (a)/
\ 4 / 1 - tan (a) \ \ 4 //
------------------ + ----------- + --------------------------------
2/ pi\ 2 / 2/ pi\\ / 2 \
1 + cot |2*a + --| 1 + tan (a) |1 + cot |a + --||*\1 + tan (a)/
\ 4 / \ \ 4 //
$$\frac{\left(- \tan^{2}{\left(a \right)} + 1\right) \left(- \cot^{2}{\left(a + \frac{\pi}{4} \right)} + 1\right)}{\left(\tan^{2}{\left(a \right)} + 1\right) \left(\cot^{2}{\left(a + \frac{\pi}{4} \right)} + 1\right)} + \frac{- \tan^{2}{\left(a \right)} + 1}{\tan^{2}{\left(a \right)} + 1} + \frac{- \cot^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1}{\cot^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1}$$
// 0 for 2*a mod pi = 0\ // 1 for a mod pi = 0\ // 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\ |< |*|< | + |< | + |< | \\sin(2*a) otherwise / \\cos(2*a) otherwise / \\sin(4*a) otherwise / \\cos(2*a) otherwise /
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for 2*a mod pi = 0\ // 0 for 4*a mod pi = 0\ || | // 1 for a mod pi = 0\ || | // 1 for a mod pi = 0\ |< / pi\ |*|< | + |< / pi\ | + |< | ||cos|2*a - --| otherwise | \\cos(2*a) otherwise / ||cos|4*a - --| otherwise | \\cos(2*a) otherwise / \\ \ 2 / / \\ \ 2 / /
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\cos{\left(2 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\cos{\left(4 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod pi = 0\ // 1 for a mod pi = 0\
// 0 for 2*a mod pi = 0\ || | // 0 for 4*a mod pi = 0\ || |
|< |*|< /pi \ | + |< | + |< /pi \ |
\\sin(2*a) otherwise / ||sin|-- + 2*a| otherwise | \\sin(4*a) otherwise / ||sin|-- + 2*a| otherwise |
\\ \2 / / \\ \2 / /
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(2 a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(2 a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
4 / 4 \
4*sin (a) 2 | 4*sin (a)|
1 - --------- 4*sin (a)*|1 - ---------|
2 2 | 2 |
sin (2*a) 4*sin (2*a) \ sin (2*a)/
------------- + -------------------------- + -------------------------
4 / 4 \ 2
4*sin (a) | 4*sin (2*a)| / 4 \
1 + --------- |1 + -----------|*sin(4*a) | 4*sin (a)|
2 | 2 | |1 + ---------| *sin(2*a)
sin (2*a) \ sin (4*a) / | 2 |
\ sin (2*a)/
$$\frac{- \frac{4 \sin^{4}{\left(a \right)}}{\sin^{2}{\left(2 a \right)}} + 1}{\frac{4 \sin^{4}{\left(a \right)}}{\sin^{2}{\left(2 a \right)}} + 1} + \frac{4 \left(- \frac{4 \sin^{4}{\left(a \right)}}{\sin^{2}{\left(2 a \right)}} + 1\right) \sin^{2}{\left(a \right)}}{\left(\frac{4 \sin^{4}{\left(a \right)}}{\sin^{2}{\left(2 a \right)}} + 1\right)^{2} \sin{\left(2 a \right)}} + \frac{4 \sin^{2}{\left(2 a \right)}}{\left(\frac{4 \sin^{4}{\left(2 a \right)}}{\sin^{2}{\left(4 a \right)}} + 1\right) \sin{\left(4 a \right)}}$$
// /pi \ \ // /pi \ \
// 0 for 2*a mod pi = 0\ || 0 for |-- + 2*a| mod pi = 0| // 0 for 4*a mod pi = 0\ || 0 for |-- + 2*a| mod pi = 0|
|< |*|< \2 / | + |< | + |< \2 / |
\\sin(2*a) otherwise / || | \\sin(4*a) otherwise / || |
\\cos(2*a) otherwise / \\cos(2*a) otherwise /
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(2 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(2 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \ // / 3*pi\ \
// 1 for a mod pi = 0\ || 1 for |2*a + ----| mod 2*pi = 0| // 1 for a mod pi = 0\ || 1 for |4*a + ----| mod 2*pi = 0|
|< |*|< \ 2 / | + |< | + |< \ 2 / |
\\cos(2*a) otherwise / || | \\cos(2*a) otherwise / || |
\\sin(2*a) otherwise / \\sin(4*a) otherwise /
$$\left(\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(2 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \left(4 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for 2*a mod pi = 0\ // 0 for 4*a mod pi = 0\ || | // 1 for a mod pi = 0\ || | // 1 for a mod pi = 0\ || 1 | || | || 1 | || | |<------------- otherwise |*|< 1 | + |<------------- otherwise | + |< 1 | || / pi\ | ||-------- otherwise | || / pi\ | ||-------- otherwise | ||sec|2*a - --| | \\sec(2*a) / ||sec|4*a - --| | \\sec(2*a) / \\ \ 2 / / \\ \ 2 / /
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{1}{\sec{\left(2 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(2 a \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{1}{\sec{\left(4 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(2 a \right)}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod pi = 0\ // 1 for a mod pi = 0\
// 0 for 2*a mod pi = 0\ || | // 0 for 4*a mod pi = 0\ || |
|| | || 1 | || | || 1 |
|< 1 |*|<------------- otherwise | + |< 1 | + |<------------- otherwise |
||-------- otherwise | || /pi \ | ||-------- otherwise | || /pi \ |
\\csc(2*a) / ||csc|-- - 2*a| | \\csc(4*a) / ||csc|-- - 2*a| |
\\ \2 / / \\ \2 / /
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{1}{\csc{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{1}{\csc{\left(4 a \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
2 / 2 \
sec (a) | sec (a) |
1 - ------------ 2*|1 - ------------|*sec(a)
2/ pi\ | 2/ pi\|
sec |a - --| | sec |a - --||
\ 2 / 2*sec(2*a) \ \ 2 //
---------------- + ---------------------------------- + -------------------------------
2 / 2 \ 2
sec (a) | sec (2*a) | / pi\ / 2 \
1 + ------------ |1 + --------------|*sec|2*a - --| | sec (a) | / pi\
2/ pi\ | 2/ pi\| \ 2 / |1 + ------------| *sec|a - --|
sec |a - --| | sec |2*a - --|| | 2/ pi\| \ 2 /
\ 2 / \ \ 2 // | sec |a - --||
\ \ 2 //
$$\frac{- \frac{\sec^{2}{\left(a \right)}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + 1}{\frac{\sec^{2}{\left(a \right)}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + 1} + \frac{2 \left(- \frac{\sec^{2}{\left(a \right)}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(a \right)}}{\left(\frac{\sec^{2}{\left(a \right)}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec{\left(a - \frac{\pi}{2} \right)}} + \frac{2 \sec{\left(2 a \right)}}{\left(\frac{\sec^{2}{\left(2 a \right)}}{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(2 a - \frac{\pi}{2} \right)}}$$
2/ pi\ / 2/ pi\\
cos |a - --| | cos |a - --||
\ 2 / | \ 2 /| / pi\
1 - ------------ / pi\ 2*|1 - ------------|*cos|a - --|
2 2*cos|2*a - --| | 2 | \ 2 /
cos (a) \ 2 / \ cos (a) /
---------------- + ----------------------------- + --------------------------------
2/ pi\ / 2/ pi\\ 2
cos |a - --| | cos |2*a - --|| / 2/ pi\\
\ 2 / | \ 2 /| | cos |a - --||
1 + ------------ |1 + --------------|*cos(2*a) | \ 2 /|
2 | 2 | |1 + ------------| *cos(a)
cos (a) \ cos (2*a) / | 2 |
\ cos (a) /
$$\frac{1 - \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(a \right)}}}{1 + \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(a \right)}}} + \frac{2 \cdot \left(1 - \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(a \right)}}\right) \cos{\left(a - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(a \right)}}\right)^{2} \cos{\left(a \right)}} + \frac{2 \cos{\left(2 a - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(2 a - \frac{\pi}{2} \right)}}{\cos^{2}{\left(2 a \right)}}\right) \cos{\left(2 a \right)}}$$
2/pi \ / 2/pi \\
csc |-- - a| | csc |-- - a||
\2 / | \2 /| /pi \
1 - ------------ /pi \ 2*|1 - ------------|*csc|-- - a|
2 2*csc|-- - 2*a| | 2 | \2 /
csc (a) \2 / \ csc (a) /
---------------- + ----------------------------- + --------------------------------
2/pi \ / 2/pi \\ 2
csc |-- - a| | csc |-- - 2*a|| / 2/pi \\
\2 / | \2 /| | csc |-- - a||
1 + ------------ |1 + --------------|*csc(2*a) | \2 /|
2 | 2 | |1 + ------------| *csc(a)
csc (a) \ csc (2*a) / | 2 |
\ csc (a) /
$$\frac{1 - \frac{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(a \right)}}}{1 + \frac{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(a \right)}}} + \frac{2 \cdot \left(1 - \frac{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(a \right)}}\right) \csc{\left(- a + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(a \right)}}\right)^{2} \csc{\left(a \right)}} + \frac{2 \csc{\left(- 2 a + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- 2 a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(2 a \right)}}\right) \csc{\left(2 a \right)}}$$
// 0 for 2*a mod pi = 0\ // 1 for a mod pi = 0\ // 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\ || | || | || | || | || 2*cot(a) | || 2 | || 2*cot(2*a) | || 2 | |<----------- otherwise |*|<-1 + cot (a) | + |<------------- otherwise | + |<-1 + cot (a) | || 2 | ||------------ otherwise | || 2 | ||------------ otherwise | ||1 + cot (a) | || 2 | ||1 + cot (2*a) | || 2 | \\ / \\1 + cot (a) / \\ / \\1 + cot (a) /
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 0 for 2*a mod pi = 0\ // 1 for a mod pi = 0\ // 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\ || | || | || | || | || 2*tan(a) | || 2 | || 2*tan(2*a) | || 2 | |<----------- otherwise |*|<1 - tan (a) | + |<------------- otherwise | + |<1 - tan (a) | || 2 | ||----------- otherwise | || 2 | ||----------- otherwise | ||1 + tan (a) | || 2 | ||1 + tan (2*a) | || 2 | \\ / \\1 + tan (a) / \\ / \\1 + tan (a) /
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \tan{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{- \tan^{2}{\left(a \right)} + 1}{\tan^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \tan{\left(2 a \right)}}{\tan^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{- \tan^{2}{\left(a \right)} + 1}{\tan^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod pi = 0\ // 1 for a mod pi = 0\
// 0 for 2*a mod pi = 0\ || | // 0 for 4*a mod pi = 0\ || |
|| | || 1 | || | || 1 |
|| 2 | ||-1 + ------- | || 2 | ||-1 + ------- |
||-------------------- otherwise | || 2 | ||------------------------ otherwise | || 2 |
|
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(a \right)}}\right) \tan{\left(a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(a \right)}}}{1 + \frac{1}{\tan^{2}{\left(a \right)}}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(2 a \right)}}\right) \tan{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(a \right)}}}{1 + \frac{1}{\tan^{2}{\left(a \right)}}} & \text{otherwise} \end{cases}\right)$$
// /pi \ \ // /pi \ \
|| 0 for |-- + 2*a| mod pi = 0| || 0 for |-- + 2*a| mod pi = 0|
// 0 for 2*a mod pi = 0\ || \2 / | // 0 for 4*a mod pi = 0\ || \2 / |
|| | || | || | || |
|| 2*cot(a) | || / pi\ | || 2*cot(2*a) | || / pi\ |
|<----------- otherwise |*|< 2*cot|a + --| | + |<------------- otherwise | + |< 2*cot|a + --| |
|| 2 | || \ 4 / | || 2 | || \ 4 / |
||1 + cot (a) | ||---------------- otherwise | ||1 + cot (2*a) | ||---------------- otherwise |
\\ / || 2/ pi\ | \\ / || 2/ pi\ |
||1 + cot |a + --| | ||1 + cot |a + --| |
\\ \ 4 / / \\ \ 4 / /
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(2 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(a + \frac{\pi}{4} \right)}}{\cot^{2}{\left(a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(2 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(a + \frac{\pi}{4} \right)}}{\cot^{2}{\left(a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \ // / 3*pi\ \
|| 1 for |2*a + ----| mod 2*pi = 0| || 1 for |4*a + ----| mod 2*pi = 0|
// 1 for a mod pi = 0\ || \ 2 / | // 1 for a mod pi = 0\ || \ 2 / |
|| | || | || | || |
|| 2 | || 2/ pi\ | || 2 | || 2/ pi\ |
|<-1 + cot (a) |*|<-1 + tan |a + --| | + |<-1 + cot (a) | + |<-1 + tan |2*a + --| |
||------------ otherwise | || \ 4 / | ||------------ otherwise | || \ 4 / |
|| 2 | ||----------------- otherwise | || 2 | ||------------------- otherwise |
\\1 + cot (a) / || 2/ pi\ | \\1 + cot (a) / || 2/ pi\ |
|| 1 + tan |a + --| | || 1 + tan |2*a + --| |
\\ \ 4 / / \\ \ 4 / /
$$\left(\left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(2 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(a + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \left(4 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(2 a + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 0 for 2*a mod pi = 0\ // 1 for a mod pi = 0\ // 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\
|| | || | || | || |
|
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\sin{\left(4 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod pi = 0\ // 1 for a mod pi = 0\
|| | || |
// 0 for 2*a mod pi = 0\ || 2 | // 0 for 4*a mod pi = 0\ || 2 |
|| | || sin (2*a) | || | || sin (2*a) |
|| sin(2*a) | ||-1 + --------- | || sin(4*a) | ||-1 + --------- |
||----------------------- otherwise | || 4 | ||--------------------------- otherwise | || 4 |
|
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{\sin{\left(2 a \right)}}{\left(1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}\right) \sin^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}}{1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{\sin{\left(4 a \right)}}{\left(1 + \frac{\sin^{2}{\left(4 a \right)}}{4 \sin^{4}{\left(2 a \right)}}\right) \sin^{2}{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}}{1 + \frac{\sin^{2}{\left(2 a \right)}}{4 \sin^{4}{\left(a \right)}}} & \text{otherwise} \end{cases}\right)$$
// 0 for 2*a mod pi = 0\ // 1 for a mod pi = 0\ // 0 for 4*a mod pi = 0\ // 1 for a mod pi = 0\ || | || | || | || | ||/ 0 for 2*a mod pi = 0 | ||/ 1 for a mod pi = 0 | ||/ 0 for 4*a mod pi = 0 | ||/ 1 for a mod pi = 0 | ||| | ||| | ||| | ||| | |<| 2*cot(a) |*|<| 2 | + |<| 2*cot(2*a) | + |<| 2 | ||<----------- otherwise otherwise | ||<-1 + cot (a) otherwise | ||<------------- otherwise otherwise | ||<-1 + cot (a) otherwise | ||| 2 | |||------------ otherwise | ||| 2 | |||------------ otherwise | |||1 + cot (a) | ||| 2 | |||1 + cot (2*a) | ||| 2 | \\\ / \\\1 + cot (a) / \\\ / \\\1 + cot (a) /
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \cot{\left(2 a \right)}}{\cot^{2}{\left(2 a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod pi = 0\ // 1 for a mod pi = 0\
|| | || |
// 0 for 2*a mod pi = 0\ || 2 | // 0 for 4*a mod pi = 0\ || 2 |
|| | || cos (a) | || | || cos (a) |
|| 2*cos(a) | ||-1 + ------------ | || 2*cos(2*a) | ||-1 + ------------ |
||------------------------------ otherwise | || 2/ pi\ | ||---------------------------------- otherwise | || 2/ pi\ |
||/ 2 \ | || cos |a - --| | ||/ 2 \ | || cos |a - --| |
|<| cos (a) | / pi\ |*|< \ 2 / | + |<| cos (2*a) | / pi\ | + |< \ 2 / |
|||1 + ------------|*cos|a - --| | ||----------------- otherwise | |||1 + --------------|*cos|2*a - --| | ||----------------- otherwise |
||| 2/ pi\| \ 2 / | || 2 | ||| 2/ pi\| \ 2 / | || 2 |
||| cos |a - --|| | || cos (a) | ||| cos |2*a - --|| | || cos (a) |
||\ \ 2 // | || 1 + ------------ | ||\ \ 2 // | || 1 + ------------ |
\\ / || 2/ pi\ | \\ / || 2/ pi\ |
|| cos |a - --| | || cos |a - --| |
\\ \ 2 / / \\ \ 2 / /
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \cos{\left(a \right)}}{\left(\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \cos{\left(2 a \right)}}{\left(\frac{\cos^{2}{\left(2 a \right)}}{\cos^{2}{\left(2 a - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(2 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(a \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod pi = 0\ // 1 for a mod pi = 0\
// 0 for 2*a mod pi = 0\ || | // 0 for 4*a mod pi = 0\ || |
|| | || 2/ pi\ | || | || 2/ pi\ |
|| / pi\ | || sec |a - --| | || / pi\ | || sec |a - --| |
|| 2*sec|a - --| | || \ 2 / | || 2*sec|2*a - --| | || \ 2 / |
|| \ 2 / | ||-1 + ------------ | || \ 2 / | ||-1 + ------------ |
||------------------------- otherwise | || 2 | ||----------------------------- otherwise | || 2 |
|
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \sec{\left(a - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}\right) \sec{\left(a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}}{1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \sec{\left(2 a - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(2 a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(2 a \right)}}\right) \sec{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}}{1 + \frac{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec^{2}{\left(a \right)}}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod pi = 0\ // 1 for a mod pi = 0\
|| | || |
// 0 for 2*a mod pi = 0\ || 2 | // 0 for 4*a mod pi = 0\ || 2 |
|| | || csc (a) | || | || csc (a) |
|| 2*csc(a) | ||-1 + ------------ | || 2*csc(2*a) | ||-1 + ------------ |
||------------------------------ otherwise | || 2/pi \ | ||---------------------------------- otherwise | || 2/pi \ |
||/ 2 \ | || csc |-- - a| | ||/ 2 \ | || csc |-- - a| |
|<| csc (a) | /pi \ |*|< \2 / | + |<| csc (2*a) | /pi \ | + |< \2 / |
|||1 + ------------|*csc|-- - a| | ||----------------- otherwise | |||1 + --------------|*csc|-- - 2*a| | ||----------------- otherwise |
||| 2/pi \| \2 / | || 2 | ||| 2/pi \| \2 / | || 2 |
||| csc |-- - a|| | || csc (a) | ||| csc |-- - 2*a|| | || csc (a) |
||\ \2 // | || 1 + ------------ | ||\ \2 // | || 1 + ------------ |
\\ / || 2/pi \ | \\ / || 2/pi \ |
|| csc |-- - a| | || csc |-- - a| |
\\ \2 / / \\ \2 / /
$$\left(\left(\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \csc{\left(a \right)}}{\left(\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 a \bmod \pi = 0 \\\frac{2 \csc{\left(2 a \right)}}{\left(\frac{\csc^{2}{\left(2 a \right)}}{\csc^{2}{\left(- 2 a + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- 2 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(a \right)}}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)$$
Piecewise((0, Mod(2*a = pi, 0)), (2*csc(a)/((1 + csc(a)^2/csc(pi/2 - a)^2)*csc(pi/2 - a)), True))*Piecewise((1, Mod(a = pi, 0)), ((-1 + csc(a)^2/csc(pi/2 - a)^2)/(1 + csc(a)^2/csc(pi/2 - a)^2), True)) + Piecewise((0, Mod(4*a = pi, 0)), (2*csc(2*a)/((1 + csc(2*a)^2/csc(pi/2 - 2*a)^2)*csc(pi/2 - 2*a)), True)) + Piecewise((1, Mod(a = pi, 0)), ((-1 + csc(a)^2/csc(pi/2 - a)^2)/(1 + csc(a)^2/csc(pi/2 - a)^2), True))
Численный ответ
[src]
cos(2*a)*sin(2*a) + cos(2*a) + sin(4*a)
cos(2*a)*sin(2*a) + cos(2*a) + sin(4*a)