Господин Экзамен
6-sin(a)^2-cos(a)^2 если a=3
Решение
Вы ввели
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2 2 6 - sin (a) - cos (a)
$$- \sin^{2}{\left(a \right)} - \cos^{2}{\left(a \right)} + 6$$
6 - sin(a)^2 - cos(a)^2
Степени
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2 2
/ I*a -I*a\ / -I*a I*a\
|e e | \- e + e /
6 - |---- + -----| + -----------------
\ 2 2 / 4
$$- \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right)^{2} + \frac{\left(e^{i a} - e^{- i a}\right)^{2}}{4} + 6$$
6 - (exp(i*a)/2 + exp(-i*a)/2)^2 + (-exp(-i*a) + exp(i*a))^2/4
Тригонометрическая часть
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5
$$5$$
2 2/ pi\
6 - sin (a) - sin |a + --|
\ 2 /
$$- \sin^{2}{\left(a \right)} - \sin^{2}{\left(a + \frac{\pi}{2} \right)} + 6$$
2 2/ pi\
6 - cos (a) - cos |a - --|
\ 2 /
$$- \cos^{2}{\left(a \right)} - \cos^{2}{\left(a - \frac{\pi}{2} \right)} + 6$$
1 1
6 - ------- - -------
2 2
csc (a) sec (a)
$$6 - \frac{1}{\sec^{2}{\left(a \right)}} - \frac{1}{\csc^{2}{\left(a \right)}}$$
1 1
6 - ------- - ------------
2 2/ pi\
sec (a) sec |a - --|
\ 2 /
$$6 - \frac{1}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} - \frac{1}{\sec^{2}{\left(a \right)}}$$
1 1
6 - ------- - ------------
2 2/pi \
sec (a) sec |-- - a|
\2 /
$$6 - \frac{1}{\sec^{2}{\left(- a + \frac{\pi}{2} \right)}} - \frac{1}{\sec^{2}{\left(a \right)}}$$
1 1
6 - ------- - ------------
2 2/pi \
csc (a) csc |-- - a|
\2 /
$$6 - \frac{1}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} - \frac{1}{\csc^{2}{\left(a \right)}}$$
1 1
6 - ------------ - ------------
2 2/pi \
csc (pi - a) csc |-- - a|
\2 /
$$6 - \frac{1}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} - \frac{1}{\csc^{2}{\left(- a + \pi \right)}}$$
7 2 cos(2*a) - + (1 - cos(a)) + 2*cos(a) - -------- 2 2
$$\left(- \cos{\left(a \right)} + 1\right)^{2} + 2 \cos{\left(a \right)} - \frac{\cos{\left(2 a \right)}}{2} + \frac{7}{2}$$
2
/ 2/a pi\\ 2
|1 - cot |- + --|| *(1 + sin(a))
1 + cos(2*a) \ \2 4 //
6 - ------------ - ---------------------------------
2 4
$$- \frac{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\sin{\left(a \right)} + 1\right)^{2}}{4} - \frac{\cos{\left(2 a \right)} + 1}{2} + 6$$
2/a pi\
4*tan |- + --|
1 - cos(2*a) \2 4 /
6 - ------------ - -------------------
2 2
/ 2/a pi\\
|1 + tan |- + --||
\ \2 4 //
$$- \frac{- \cos{\left(2 a \right)} + 1}{2} + 6 - \frac{4 \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2
/ 2/a\\ 2/a\
|1 - tan |-|| 4*tan |-|
\ \2// \2/
6 - -------------- - --------------
2 2
/ 2/a\\ / 2/a\\
|1 + tan |-|| |1 + tan |-||
\ \2// \ \2//
$$- \frac{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} + 6 - \frac{4 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
2/a\ 2/a pi\
4*tan |-| 4*tan |- + --|
\2/ \2 4 /
6 - -------------- - -------------------
2 2
/ 2/a\\ / 2/a pi\\
|1 + tan |-|| |1 + tan |- + --||
\ \2// \ \2 4 //
$$6 - \frac{4 \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} - \frac{4 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
2/a\ 2/a pi\
4*cot |-| 4*tan |- + --|
\2/ \2 4 /
6 - -------------- - -------------------
2 2
/ 2/a\\ / 2/a pi\\
|1 + cot |-|| |1 + tan |- + --||
\ \2// \ \2 4 //
$$6 - \frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} - \frac{4 \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
6 - |< 2 | - |< 2 |
||sin (a) otherwise | ||cos (a) otherwise |
\\ / \\ /
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right) + 6$$
2
/ 1 \
|1 - -------|
| 2/a\|
| cot |-||
\ \2// 4
6 - -------------- - ----------------------
2 2
/ 1 \ / 1 \ 2/a\
|1 + -------| |1 + -------| *cot |-|
| 2/a\| | 2/a\| \2/
| cot |-|| | cot |-||
\ \2// \ \2//
$$- \frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)^{2}} + 6 - \frac{4}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \cot^{2}{\left(\frac{a}{2} \right)}}$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
6 - |< 2/ pi\ | - |< 2 |
||cos |a - --| otherwise | ||cos (a) otherwise |
\\ \ 2 / / \\ /
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\cos^{2}{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right) + 6$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
6 - |< 2 | - |< 2/ pi\ |
||sin (a) otherwise | ||sin |a + --| otherwise |
\\ / \\ \ 2 / /
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\sin^{2}{\left(a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + 6$$
2 2
/ 2/a\\ / 2/a pi\\
|-1 + cot |-|| |-1 + tan |- + --||
\ \2// \ \2 4 //
6 - --------------- - --------------------
2 2
/ 2/a\\ / 2/a pi\\
|1 + cot |-|| |1 + tan |- + --||
\ \2// \ \2 4 //
$$- \frac{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} - \frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} + 6$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
|| 1 | || 1 |
6 - |<------------ otherwise | - |<------- otherwise |
|| 2/ pi\ | || 2 |
||sec |a - --| | ||sec (a) |
\\ \ 2 / / \\ /
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right) + 6$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
|| 1 | || 1 |
6 - |<------- otherwise | - |<------------ otherwise |
|| 2 | || 2/pi \ |
||csc (a) | ||csc |-- - a| |
\\ / \\ \2 / /
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 6$$
2 2
/ 2/a pi\\ / 2/a\\
|1 - cot |- + --|| |1 - tan |-||
\ \2 4 // \ \2//
6 - ------------------- - --------------
2 2
/ 2/a pi\\ / 2/a\\
|1 + cot |- + --|| |1 + tan |-||
\ \2 4 // \ \2//
$$- \frac{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} - \frac{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + 6$$
// / 3*pi\ \
// 1 for a mod 2*pi = 0\ || 1 for |a + ----| mod 2*pi = 0|
|| | || \ 2 / |
6 - |< 2 | - |< |
||cos (a) otherwise | || 4/a\ 2/a\ |
\\ / ||- 4*cos |-| + 4*cos |-| otherwise |
\\ \2/ \2/ /
$$\left(- \begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\- 4 \cos^{4}{\left(\frac{a}{2} \right)} + 4 \cos^{2}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right) + 6$$
// / pi\ \
// 0 for a mod pi = 0\ || 0 for |a + --| mod pi = 0|
|| | || \ 2 / |
6 - |< 2 | - |< |
||sin (a) otherwise | || 2 2/a pi\ |
\\ / ||(1 + sin(a)) *cot |- + --| otherwise |
\\ \2 4 / /
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(a \right)} + 1\right)^{2} \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right) + 6$$
2
/ 4/a\\
| 4*sin |-||
| \2/|
|1 - ---------| 4/a\
| 2 | 16*sin |-|
\ sin (a) / \2/
6 - ---------------- - ------------------------
2 2
/ 4/a\\ / 4/a\\
| 4*sin |-|| | 4*sin |-||
| \2/| | \2/| 2
|1 + ---------| |1 + ---------| *sin (a)
| 2 | | 2 |
\ sin (a) / \ sin (a) /
$$- \frac{\left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{2}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{2}} + 6 - \frac{16 \sin^{4}{\left(\frac{a}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{2} \sin^{2}{\left(a \right)}}$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
||/ 0 for a mod pi = 0 | ||/ 1 for a mod 2*pi = 0 |
6 - |<| | - |<| |
||< 2 otherwise | ||< 2 otherwise |
|||sin (a) otherwise | |||cos (a) otherwise |
\\\ / \\\ /
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 6$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
|| 2/a\ | || 2 |
|| 4*cot |-| | ||/ 2/a\\ |
|| \2/ | |||-1 + cot |-|| |
6 - |<-------------- otherwise | - |<\ \2// |
|| 2 | ||--------------- otherwise |
||/ 2/a\\ | || 2 |
|||1 + cot |-|| | || / 2/a\\ |
||\ \2// | || |1 + cot |-|| |
\\ / \\ \ \2// /
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + 6$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
|| 2/a\ | || 2 |
|| 4*tan |-| | ||/ 2/a\\ |
|| \2/ | |||1 - tan |-|| |
6 - |<-------------- otherwise | - |<\ \2// |
|| 2 | ||-------------- otherwise |
||/ 2/a\\ | || 2 |
|||1 + tan |-|| | ||/ 2/a\\ |
||\ \2// | |||1 + tan |-|| |
\\ / \\\ \2// /
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + 6$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 2 |
|| | ||/ 1 \ |
|| 4 | |||-1 + -------| |
||---------------------- otherwise | ||| 2/a\| |
|| 2 | ||| tan |-|| |
6 - |
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \tan^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right) + 6$$
// / pi\ \
// 0 for a mod pi = 0\ || 0 for |a + --| mod pi = 0|
|| | || \ 2 / |
|| 2/a\ | || |
|| 4*cot |-| | || 2/a pi\ |
|| \2/ | || 4*cot |- + --| |
6 - |<-------------- otherwise | - |< \2 4 / |
|| 2 | ||------------------- otherwise |
||/ 2/a\\ | || 2 |
|||1 + cot |-|| | ||/ 2/a pi\\ |
||\ \2// | |||1 + cot |- + --|| |
\\ / ||\ \2 4 // |
\\ /
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + 6$$
2
/ 2/a pi\\
| cos |- - --||
| \2 2 /|
|1 - ------------|
| 2/a\ | 2/a pi\
| cos |-| | 4*cos |- - --|
\ \2/ / \2 2 /
6 - ------------------- - ---------------------------
2 2
/ 2/a pi\\ / 2/a pi\\
| cos |- - --|| | cos |- - --||
| \2 2 /| | \2 2 /| 2/a\
|1 + ------------| |1 + ------------| *cos |-|
| 2/a\ | | 2/a\ | \2/
| cos |-| | | cos |-| |
\ \2/ / \ \2/ /
$$- \frac{\left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)^{2}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)^{2}} + 6 - \frac{4 \cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \cos^{2}{\left(\frac{a}{2} \right)}}$$
2
/ 2/a\ \
| sec |-| |
| \2/ |
|1 - ------------|
| 2/a pi\| 2/a\
| sec |- - --|| 4*sec |-|
\ \2 2 // \2/
6 - ------------------- - --------------------------------
2 2
/ 2/a\ \ / 2/a\ \
| sec |-| | | sec |-| |
| \2/ | | \2/ | 2/a pi\
|1 + ------------| |1 + ------------| *sec |- - --|
| 2/a pi\| | 2/a pi\| \2 2 /
| sec |- - --|| | sec |- - --||
\ \2 2 // \ \2 2 //
$$- \frac{\left(- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} + 6 - \frac{4 \sec^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}$$
2
/ 2/pi a\\
| csc |-- - -||
| \2 2/|
|1 - ------------|
| 2/a\ | 2/pi a\
| csc |-| | 4*csc |-- - -|
\ \2/ / \2 2/
6 - ------------------- - ---------------------------
2 2
/ 2/pi a\\ / 2/pi a\\
| csc |-- - -|| | csc |-- - -||
| \2 2/| | \2 2/| 2/a\
|1 + ------------| |1 + ------------| *csc |-|
| 2/a\ | | 2/a\ | \2/
| csc |-| | | csc |-| |
\ \2/ / \ \2/ /
$$- \frac{\left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)^{2}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)^{2}} + 6 - \frac{4 \csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \csc^{2}{\left(\frac{a}{2} \right)}}$$
// / 3*pi\ \
// 1 for a mod 2*pi = 0\ || 1 for |a + ----| mod 2*pi = 0|
|| | || \ 2 / |
|| 2 | || |
||/ 2/a\\ | || 2 |
|||-1 + cot |-|| | ||/ 2/a pi\\ |
6 - |<\ \2// | - |<|-1 + tan |- + --|| |
||--------------- otherwise | ||\ \2 4 // |
|| 2 | ||-------------------- otherwise |
|| / 2/a\\ | || 2 |
|| |1 + cot |-|| | ||/ 2/a pi\\ |
\\ \ \2// / |||1 + tan |- + --|| |
\\\ \2 4 // /
$$\left(- \begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + 6$$
// 0 for a mod pi = 0\
|| | // 1 for a mod 2*pi = 0\
|| 2 | || |
|| sin (a) | || 2 |
||------------------------ otherwise | ||/ 2 4/a\\ |
|| 2 | |||sin (a) - 4*sin |-|| |
6 - |
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin^{2}{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}\right)^{2}}{\left(4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}\right)^{2}} & \text{otherwise} \end{cases}\right) + 6$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 2 |
|| | ||/ 2 \ |
|| 2 | ||| sin (a) | |
|| sin (a) | |||-1 + ---------| |
||------------------------ otherwise | ||| 4/a\| |
|| 2 | ||| 4*sin |-|| |
6 - |
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin^{2}{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right) + 6$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
||/ 0 for a mod pi = 0 | ||/ 1 for a mod 2*pi = 0 |
||| | ||| |
||| 2/a\ | ||| 2 |
||| 4*cot |-| | |||/ 2/a\\ |
6 - |<| \2/ | - |<||-1 + cot |-|| |
||<-------------- otherwise otherwise | ||<\ \2// otherwise |
||| 2 | |||--------------- otherwise |
|||/ 2/a\\ | ||| 2 |
||||1 + cot |-|| | ||| / 2/a\\ |
|||\ \2// | ||| |1 + cot |-|| |
\\\ / \\\ \ \2// /
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 6$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 2 |
|| | ||/ 2/a\ \ |
|| 2/a\ | ||| cos |-| | |
|| 4*cos |-| | ||| \2/ | |
|| \2/ | |||-1 + ------------| |
||-------------------------------- otherwise | ||| 2/a pi\| |
|| 2 | ||| cos |- - --|| |
6 - |
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \cos^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + 6$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 2 |
|| | ||/ 2/a pi\\ |
|| 2/a pi\ | ||| sec |- - --|| |
|| 4*sec |- - --| | ||| \2 2 /| |
|| \2 2 / | |||-1 + ------------| |
||--------------------------- otherwise | ||| 2/a\ | |
|| 2 | ||| sec |-| | |
6 - |
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \sec^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right) + 6$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 2 |
|| | ||/ 2/a\ \ |
|| 2/a\ | ||| csc |-| | |
|| 4*csc |-| | ||| \2/ | |
|| \2/ | |||-1 + ------------| |
||-------------------------------- otherwise | ||| 2/pi a\| |
|| 2 | ||| csc |-- - -|| |
6 - |
$$\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \csc^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2} \csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) + 6$$
6 - Piecewise((0, Mod(a = pi, 0)), (4*csc(a/2)^2/((1 + csc(a/2)^2/csc(pi/2 - a/2)^2)^2*csc(pi/2 - a/2)^2), True)) - Piecewise((1, Mod(a = 2*pi, 0)), ((-1 + csc(a/2)^2/csc(pi/2 - a/2)^2)^2/(1 + csc(a/2)^2/csc(pi/2 - a/2)^2)^2, True))