Господин Экзамен
cos(factorial(n)) если n=1
Решение
Подстановка условия
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cos(factorial(n)) при n = 1
подставляем
cos(n!)
$$\cos{\left(n! \right)}$$
cos(n!)
$$\cos{\left(n! \right)}$$
переменные
n = 1
$$n = 1$$
cos((1)!)
$$\cos{\left((1)! \right)}$$
cos(1!)
$$\cos{\left(1! \right)}$$
cos(1)
$$\cos{\left(1 \right)}$$
cos(1)
Комбинаторика
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cos(Gamma(1 + n))
$$\cos{\left(\Gamma\left(n + 1\right) \right)}$$
cos(gamma(1 + n))
Степени
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I*n! -I*n! e e ----- + ------ 2 2
$$\frac{e^{i n!}}{2} + \frac{e^{- i n!}}{2}$$
exp(i*factorial(n))/2 + exp(-i*factorial(n))/2
Тригонометрическая часть
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1 ------- sec(n!)
$$\frac{1}{\sec{\left(n! \right)}}$$
/pi \ sin|-- + n!| \2 /
$$\sin{\left(n! + \frac{\pi}{2} \right)}$$
1 ------------ /pi \ csc|-- - n!| \2 /
$$\frac{1}{\csc{\left(- n! + \frac{\pi}{2} \right)}}$$
2/n!\
-1 + cot |--|
\2 /
-------------
2/n!\
1 + cot |--|
\2 /
$$\frac{\cot^{2}{\left(\frac{n!}{2} \right)} - 1}{\cot^{2}{\left(\frac{n!}{2} \right)} + 1}$$
2/n!\
1 - tan |--|
\2 /
------------
2/n!\
1 + tan |--|
\2 /
$$\frac{- \tan^{2}{\left(\frac{n!}{2} \right)} + 1}{\tan^{2}{\left(\frac{n!}{2} \right)} + 1}$$
1
1 - --------
2/n!\
cot |--|
\2 /
------------
1
1 + --------
2/n!\
cot |--|
\2 /
$$\frac{1 - \frac{1}{\cot^{2}{\left(\frac{n!}{2} \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{n!}{2} \right)}}}$$
/n! pi\
2*tan|-- + --|
\2 4 /
-----------------
2/n! pi\
1 + tan |-- + --|
\2 4 /
$$\frac{2 \tan{\left(\frac{n!}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{n!}{2} + \frac{\pi}{4} \right)} + 1}$$
/ 1 for And(im(n!) = 0, n! mod 2*pi = 0) < \cos(n!) otherwise
$$\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(n!\right)} = 0 \wedge n! \bmod 2 \pi = 0 \\\cos{\left(n! \right)} & \text{otherwise} \end{cases}$$
/ 1 for And(im(n!) = 0, n! mod 2*pi = 0) | < 1 |------- otherwise \sec(n!)
$$\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(n!\right)} = 0 \wedge n! \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(n! \right)}} & \text{otherwise} \end{cases}$$
2*(-1 - cos(2*n!) + 2*cos(n!))
--------------------------------
2
1 - cos(2*n!) + 2*(1 - cos(n!))
$$\frac{2 \cdot \left(2 \cos{\left(n! \right)} - \cos{\left(2 n! \right)} - 1\right)}{2 \left(- \cos{\left(n! \right)} + 1\right)^{2} - \cos{\left(2 n! \right)} + 1}$$
4/n!\
4*sin |--|
\2 /
1 - ----------
2
sin (n!)
--------------
4/n!\
4*sin |--|
\2 /
1 + ----------
2
sin (n!)
$$\frac{- \frac{4 \sin^{4}{\left(\frac{n!}{2} \right)}}{\sin^{2}{\left(n! \right)}} + 1}{\frac{4 \sin^{4}{\left(\frac{n!}{2} \right)}}{\sin^{2}{\left(n! \right)}} + 1}$$
/ 1 for And(im(n!) = 0, n! mod 2*pi = 0) | < /pi \ |sin|-- + n!| otherwise \ \2 /
$$\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(n!\right)} = 0 \wedge n! \bmod 2 \pi = 0 \\\sin{\left(n! + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}$$
/ 1 for And(im(n!) = 0, n! mod 2*pi = 0) | | 1 <------------ otherwise | /pi \ |csc|-- - n!| \ \2 /
$$\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(n!\right)} = 0 \wedge n! \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- n! + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}$$
2/n!\
sec |--|
\2 /
1 - -------------
2/n! pi\
sec |-- - --|
\2 2 /
-----------------
2/n!\
sec |--|
\2 /
1 + -------------
2/n! pi\
sec |-- - --|
\2 2 /
$$\frac{1 - \frac{\sec^{2}{\left(\frac{n!}{2} \right)}}{\sec^{2}{\left(\frac{n!}{2} - \frac{\pi}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{n!}{2} \right)}}{\sec^{2}{\left(\frac{n!}{2} - \frac{\pi}{2} \right)}}}$$
2/n! pi\
cos |-- - --|
\2 2 /
1 - -------------
2/n!\
cos |--|
\2 /
-----------------
2/n! pi\
cos |-- - --|
\2 2 /
1 + -------------
2/n!\
cos |--|
\2 /
$$\frac{- \frac{\cos^{2}{\left(\frac{n!}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{n!}{2} \right)}} + 1}{\frac{\cos^{2}{\left(\frac{n!}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{n!}{2} \right)}} + 1}$$
2/pi n!\
csc |-- - --|
\2 2 /
1 - -------------
2/n!\
csc |--|
\2 /
-----------------
2/pi n!\
csc |-- - --|
\2 2 /
1 + -------------
2/n!\
csc |--|
\2 /
$$\frac{- \frac{\csc^{2}{\left(- \frac{n!}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{n!}{2} \right)}} + 1}{\frac{\csc^{2}{\left(- \frac{n!}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{n!}{2} \right)}} + 1}$$
/ 1 for And(im(n!) = 0, n! mod 2*pi = 0) | | 2/n!\ |-1 + cot |--| < \2 / |------------- otherwise | 2/n!\ | 1 + cot |--| \ \2 /
$$\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(n!\right)} = 0 \wedge n! \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{n!}{2} \right)} - 1}{\cot^{2}{\left(\frac{n!}{2} \right)} + 1} & \text{otherwise} \end{cases}$$
/ 1 for And(im(n!) = 0, n! mod 2*pi = 0) | | 2/n!\ |1 - tan |--| < \2 / |------------ otherwise | 2/n!\ |1 + tan |--| \ \2 /
$$\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(n!\right)} = 0 \wedge n! \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{n!}{2} \right)} + 1}{\tan^{2}{\left(\frac{n!}{2} \right)} + 1} & \text{otherwise} \end{cases}$$
/ 1 for And(im(n!) = 0, n! mod 2*pi = 0) | | 1 |-1 + -------- | 2/n!\ | tan |--| < \2 / |------------- otherwise | 1 | 1 + -------- | 2/n!\ | tan |--| \ \2 /
$$\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(n!\right)} = 0 \wedge n! \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{n!}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{n!}{2} \right)}}} & \text{otherwise} \end{cases}$$
/ / /pi \ \ | 0 for And|im(n!) = 0, |-- + n!| mod pi = 0| | \ \2 / / < | /n! pi\ |(1 + sin(n!))*cot|-- + --| otherwise \ \2 4 /
$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(n!\right)} = 0 \wedge \left(n! + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(n! \right)} + 1\right) \cot{\left(\frac{n!}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}$$
/ / /pi \ \ | 0 for And|im(n!) = 0, |-- + n!| mod pi = 0| | \ \2 / / | | /n! pi\ < 2*cot|-- + --| | \2 4 / |----------------- otherwise | 2/n! pi\ |1 + cot |-- + --| \ \2 4 /
$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(n!\right)} = 0 \wedge \left(n! + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{n!}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{n!}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}$$
/ 1 for And(im(n!) = 0, n! mod 2*pi = 0) | | 2 < -4 + 4*sin (n!) + 4*cos(n!) |----------------------------- otherwise | 2 2 \2*(1 - cos(n!)) + 2*sin (n!)
$$\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(n!\right)} = 0 \wedge n! \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(n! \right)} + 4 \cos{\left(n! \right)} - 4}{2 \left(- \cos{\left(n! \right)} + 1\right)^{2} + 2 \sin^{2}{\left(n! \right)}} & \text{otherwise} \end{cases}$$
/ 1 for And(im(n!) = 0, n! mod 2*pi = 0)
|
$$\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(n!\right)} = 0 \wedge n! \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(n!\right)} = 0 \wedge n! \bmod 2 \pi = 0 \\\cos{\left(n! \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}$$
/ 1 for And(im(n!) = 0, n! mod 2*pi = 0) | | 2 | sin (n!) |-1 + ---------- | 4/n!\ | 4*sin |--| < \2 / |--------------- otherwise | 2 | sin (n!) | 1 + ---------- | 4/n!\ | 4*sin |--| \ \2 /
$$\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(n!\right)} = 0 \wedge n! \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(n! \right)}}{4 \sin^{4}{\left(\frac{n!}{2} \right)}}}{1 + \frac{\sin^{2}{\left(n! \right)}}{4 \sin^{4}{\left(\frac{n!}{2} \right)}}} & \text{otherwise} \end{cases}$$
/ 1 for And(im(n!) = 0, n! mod 2*pi = 0) | | 2/n! pi\ | sec |-- - --| | \2 2 / |-1 + ------------- | 2/n!\ | sec |--| < \2 / |------------------ otherwise | 2/n! pi\ | sec |-- - --| | \2 2 / |1 + ------------- | 2/n!\ | sec |--| \ \2 /
$$\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(n!\right)} = 0 \wedge n! \bmod 2 \pi = 0 \\\frac{\frac{\sec^{2}{\left(\frac{n!}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{n!}{2} \right)}} - 1}{\frac{\sec^{2}{\left(\frac{n!}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{n!}{2} \right)}} + 1} & \text{otherwise} \end{cases}$$
/ 1 for And(im(n!) = 0, n! mod 2*pi = 0) | | 2/n!\ | cos |--| | \2 / |-1 + ------------- | 2/n! pi\ | cos |-- - --| < \2 2 / |------------------ otherwise | 2/n!\ | cos |--| | \2 / |1 + ------------- | 2/n! pi\ | cos |-- - --| \ \2 2 /
$$\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(n!\right)} = 0 \wedge n! \bmod 2 \pi = 0 \\\frac{-1 + \frac{\cos^{2}{\left(\frac{n!}{2} \right)}}{\cos^{2}{\left(\frac{n!}{2} - \frac{\pi}{2} \right)}}}{1 + \frac{\cos^{2}{\left(\frac{n!}{2} \right)}}{\cos^{2}{\left(\frac{n!}{2} - \frac{\pi}{2} \right)}}} & \text{otherwise} \end{cases}$$
/ 1 for And(im(n!) = 0, n! mod 2*pi = 0) | | 2/n!\ | csc |--| | \2 / |-1 + ------------- | 2/pi n!\ | csc |-- - --| < \2 2 / |------------------ otherwise | 2/n!\ | csc |--| | \2 / |1 + ------------- | 2/pi n!\ | csc |-- - --| \ \2 2 /
$$\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(n!\right)} = 0 \wedge n! \bmod 2 \pi = 0 \\\frac{-1 + \frac{\csc^{2}{\left(\frac{n!}{2} \right)}}{\csc^{2}{\left(- \frac{n!}{2} + \frac{\pi}{2} \right)}}}{1 + \frac{\csc^{2}{\left(\frac{n!}{2} \right)}}{\csc^{2}{\left(- \frac{n!}{2} + \frac{\pi}{2} \right)}}} & \text{otherwise} \end{cases}$$
/ 1 for And(im(n!) = 0, n! mod 2*pi = 0) | |/ 1 for And(im(n!) = 0, n! mod 2*pi = 0) || || 2/n!\ <|-1 + cot |--| |< \2 / otherwise ||------------- otherwise || 2/n!\ || 1 + cot |--| \\ \2 /
$$\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(n!\right)} = 0 \wedge n! \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(n!\right)} = 0 \wedge n! \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{n!}{2} \right)} - 1}{\cot^{2}{\left(\frac{n!}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}$$
Piecewise((1, (im(factorial(n)) = 0))∧(Eq(Mod(factorial(n, 2*pi), 0))), (Piecewise((1, (im(factorial(n)) = 0))∧(Eq(Mod(factorial(n, 2*pi), 0))), ((-1 + cot(factorial(n)/2)^2)/(1 + cot(factorial(n)/2)^2), True)), True))