Господин Экзамен
cos(a)-1 если a=4
Решение
Подстановка условия
[src]
cos(a) - 1*1 при a = 4
подставляем
cos(a) - 1
$$\cos{\left(a \right)} - 1$$
-1 + cos(a)
$$\cos{\left(a \right)} - 1$$
переменные
a = 4
$$a = 4$$
-1 + cos((4))
$$\cos{\left((4) \right)} - 1$$
-1 + cos(4)
$$-1 + \cos{\left(4 \right)}$$
-1 + cos(4)
Степени
[src]
I*a -I*a
e e
-1 + ---- + -----
2 2
$$\frac{e^{i a}}{2} - 1 + \frac{e^{- i a}}{2}$$
-1 + exp(i*a)/2 + exp(-i*a)/2
Тригонометрическая часть
[src]
1
-1 + ------
sec(a)
$$-1 + \frac{1}{\sec{\left(a \right)}}$$
/ pi\
-1 + sin|a + --|
\ 2 /
$$\sin{\left(a + \frac{\pi}{2} \right)} - 1$$
1
-1 + -----------
/pi \
csc|-- - a|
\2 /
$$-1 + \frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}}$$
2
1 + cos (a) - 2*cos(a)
----------------------
-1 + cos(a)
$$\frac{\cos^{2}{\left(a \right)} - 2 \cos{\left(a \right)} + 1}{\cos{\left(a \right)} - 1}$$
2/a\
-1 + cot |-|
\2/
-1 + ------------
2/a\
1 + cot |-|
\2/
$$\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} - 1$$
2/a\
1 - tan |-|
\2/
-1 + -----------
2/a\
1 + tan |-|
\2/
$$\frac{- \tan^{2}{\left(\frac{a}{2} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} - 1$$
1
1 - -------
2/a\
cot |-|
\2/
-1 + -----------
1
1 + -------
2/a\
cot |-|
\2/
$$\frac{1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}} - 1$$
/a pi\
2*tan|- + --|
\2 4 /
-1 + ----------------
2/a pi\
1 + tan |- + --|
\2 4 /
$$-1 + \frac{2 \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}$$
// 1 for a mod 2*pi = 0\
-1 + |< |
\\cos(a) otherwise /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) - 1$$
// 1 for a mod 2*pi = 0\
|| |
-1 + |< 1 |
||------ otherwise |
\\sec(a) /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(a \right)}} & \text{otherwise} \end{cases}\right) - 1$$
// 1 for a mod 2*pi = 0\
|| |
-1 + |< / pi\ |
||sin|a + --| otherwise |
\\ \ 2 / /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\sin{\left(a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) - 1$$
// 1 for a mod 2*pi = 0\
|| |
|| 1 |
-1 + |<----------- otherwise |
|| /pi \ |
||csc|-- - a| |
\\ \2 / /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - 1$$
4/a\
4*sin |-|
\2/
1 - ---------
2
sin (a)
-1 + -------------
4/a\
4*sin |-|
\2/
1 + ---------
2
sin (a)
$$\frac{- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1}{\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1} - 1$$
// 1 for a mod 2*pi = 0\
|| |
|| 2/a\ |
||-1 + cot |-| |
-1 + |< \2/ |
||------------ otherwise |
|| 2/a\ |
||1 + cot |-| |
\\ \2/ /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) - 1$$
// 1 for a mod 2*pi = 0\
|| |
|| 2/a\ |
||1 - tan |-| |
-1 + |< \2/ |
||----------- otherwise |
|| 2/a\ |
||1 + tan |-| |
\\ \2/ /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{a}{2} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) - 1$$
// 1 for a mod 2*pi = 0\
|| |
-1 + |
$$\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - 1$$
// 1 for a mod 2*pi = 0\
|| |
|| 1 |
||-1 + ------- |
|| 2/a\ |
|| tan |-| |
-1 + |< \2/ |
||------------ otherwise |
|| 1 |
||1 + ------- |
|| 2/a\ |
|| tan |-| |
\\ \2/ /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) - 1$$
2/a pi\
cos |- - --|
\2 2 /
1 - ------------
2/a\
cos |-|
\2/
-1 + ----------------
2/a pi\
cos |- - --|
\2 2 /
1 + ------------
2/a\
cos |-|
\2/
$$\frac{1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}} - 1$$
// / pi\ \
|| 0 for |a + --| mod pi = 0|
|| \ 2 / |
-1 + |< |
|| /a pi\ |
||(1 + sin(a))*cot|- + --| otherwise |
\\ \2 4 / /
$$\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(a \right)} + 1\right) \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right) - 1$$
2/a\
sec |-|
\2/
1 - ------------
2/a pi\
sec |- - --|
\2 2 /
-1 + ----------------
2/a\
sec |-|
\2/
1 + ------------
2/a pi\
sec |- - --|
\2 2 /
$$\frac{- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1}{\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1} - 1$$
2/pi a\
csc |-- - -|
\2 2/
1 - ------------
2/a\
csc |-|
\2/
-1 + ----------------
2/pi a\
csc |-- - -|
\2 2/
1 + ------------
2/a\
csc |-|
\2/
$$\frac{1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}} - 1$$
// / pi\ \
|| 0 for |a + --| mod pi = 0|
|| \ 2 / |
|| |
|| /a pi\ |
-1 + |< 2*cot|- + --| |
|| \2 4 / |
||---------------- otherwise |
|| 2/a pi\ |
||1 + cot |- + --| |
\\ \2 4 / /
$$\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) - 1$$
// 1 for a mod 2*pi = 0\
|| |
|| 2 |
-1 + |< -4 + 4*sin (a) + 4*cos(a) |
||--------------------------- otherwise |
|| 2 2 |
\\2*(1 - cos(a)) + 2*sin (a) /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(a \right)} + 4 \cos{\left(a \right)} - 4}{2 \left(- \cos{\left(a \right)} + 1\right)^{2} + 2 \sin^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right) - 1$$
// 1 for a mod 2*pi = 0\
|| |
|| 2 |
|| sin (a) |
||-1 + --------- |
|| 4/a\ |
|| 4*sin |-| |
-1 + |< \2/ |
||-------------- otherwise |
|| 2 |
|| sin (a) |
||1 + --------- |
|| 4/a\ |
|| 4*sin |-| |
\\ \2/ /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) - 1$$
// 1 for a mod 2*pi = 0\
|| |
||/ 1 for a mod 2*pi = 0 |
||| |
||| 2/a\ |
-1 + |<|-1 + cot |-| |
||< \2/ otherwise |
|||------------ otherwise |
||| 2/a\ |
|||1 + cot |-| |
\\\ \2/ /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - 1$$
// 1 for a mod 2*pi = 0\
|| |
|| 2/a\ |
|| cos |-| |
|| \2/ |
||-1 + ------------ |
|| 2/a pi\ |
|| cos |- - --| |
-1 + |< \2 2 / |
||----------------- otherwise |
|| 2/a\ |
|| cos |-| |
|| \2/ |
|| 1 + ------------ |
|| 2/a pi\ |
|| cos |- - --| |
\\ \2 2 / /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) - 1$$
// 1 for a mod 2*pi = 0\
|| |
|| 2/a pi\ |
|| sec |- - --| |
|| \2 2 / |
||-1 + ------------ |
|| 2/a\ |
|| sec |-| |
-1 + |< \2/ |
||----------------- otherwise |
|| 2/a pi\ |
|| sec |- - --| |
|| \2 2 / |
|| 1 + ------------ |
|| 2/a\ |
|| sec |-| |
\\ \2/ /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) - 1$$
// 1 for a mod 2*pi = 0\
|| |
|| 2/a\ |
|| csc |-| |
|| \2/ |
||-1 + ------------ |
|| 2/pi a\ |
|| csc |-- - -| |
-1 + |< \2 2/ |
||----------------- otherwise |
|| 2/a\ |
|| csc |-| |
|| \2/ |
|| 1 + ------------ |
|| 2/pi a\ |
|| csc |-- - -| |
\\ \2 2/ /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) - 1$$
-1 + Piecewise((1, Mod(a = 2*pi, 0)), ((-1 + csc(a/2)^2/csc(pi/2 - a/2)^2)/(1 + csc(a/2)^2/csc(pi/2 - a/2)^2), True))