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Упрощение выражений/ Общий знаменатель/ 1-2*sin(a)^(2)/2*cot(pi/4+a)*cos(pi/4-a)^(2)

Общий знаменатель 1-2*sin(a)^(2)/2*cot(pi/4+a)*cos(pi/4-a)^(2)

Выражение, которое надо упростить:

Решение

Вы ввели [src] 
         2       /pi    \    2/pi    \
    2*sin (a)*cot|-- + a|*cos |-- - a|
                 \4     /     \4     /
1 - ----------------------------------
                    2
$$- \frac{2 \sin^{2}{\left(a \right)} \cos^{2}{\left(- a + \frac{\pi}{4} \right)} \cot{\left(a + \frac{\pi}{4} \right)}}{2} + 1$$ 
1 - 2*sin(a)^2*cot(pi/4 + a)*cos(pi/4 - a)^2/2
Общее упрощение [src] 
                 2   
       4      sin (a)
1 + sin (a) - -------
                 2
$$\sin^{4}{\left(a \right)} - \frac{\sin^{2}{\left(a \right)}}{2} + 1$$ 
1 + sin(a)^4 - sin(a)^2/2
Собрать выражение [src] 
       2/pi    \    2       /pi    \
1 - cos |-- - a|*sin (a)*cot|-- + a|
        \4     /            \4     /
$$- \sin^{2}{\left(a \right)} \cos^{2}{\left(- a + \frac{\pi}{4} \right)} \cot{\left(a + \frac{\pi}{4} \right)} + 1$$ 
    /1   cos(2*a)   sin(4*a)   sin(2*a)\    /    pi\
1 - |- - -------- - -------- + --------|*cot|a + --|
    \4      4          8          4    /    \    4 /
$$- \left(\frac{\sin{\left(2 a \right)}}{4} - \frac{\sin{\left(4 a \right)}}{8} - \frac{\cos{\left(2 a \right)}}{4} + \frac{1}{4}\right) \cot{\left(a + \frac{\pi}{4} \right)} + 1$$ 
1 - (1/4 - cos(2*a)/4 - sin(4*a)/8 + sin(2*a)/4)*cot(a + pi/4)
Общий знаменатель [src] 
       2       2/    pi\    /    pi\
1 - sin (a)*sin |a + --|*cot|a + --|
                \    4 /    \    4 /
$$- \sin^{2}{\left(a \right)} \sin^{2}{\left(a + \frac{\pi}{4} \right)} \cot{\left(a + \frac{\pi}{4} \right)} + 1$$ 
1 - sin(a)^2*sin(a + pi/4)^2*cot(a + pi/4)
Раскрыть выражение [src] 
                                 2                      
    /  ___            ___       \                       
    |\/ 2 *cos(a)   \/ 2 *sin(a)|     2                 
    |------------ + ------------| *sin (a)*(-1 + cot(a))
    \     2              2      /                       
1 - ----------------------------------------------------
                         1 + cot(a)
$$- \frac{\left(\frac{\sqrt{2} \sin{\left(a \right)}}{2} + \frac{\sqrt{2} \cos{\left(a \right)}}{2}\right)^{2} \left(\cot{\left(a \right)} - 1\right) \sin^{2}{\left(a \right)}}{\cot{\left(a \right)} + 1} + 1$$ 
         4            3                2       2         4                3                       2       2          
      sin (a)      sin (a)*cos(a)   cos (a)*sin (a)   sin (a)*cot(a)   sin (a)*cos(a)*cot(a)   cos (a)*sin (a)*cot(a)
1 + ------------ + -------------- + --------------- - -------------- - --------------------- - ----------------------
    2 + 2*cot(a)     1 + cot(a)       2 + 2*cot(a)     2 + 2*cot(a)          1 + cot(a)             2 + 2*cot(a)
$$- \frac{\sin^{4}{\left(a \right)} \cot{\left(a \right)}}{2 \cot{\left(a \right)} + 2} - \frac{\sin^{2}{\left(a \right)} \cos^{2}{\left(a \right)} \cot{\left(a \right)}}{2 \cot{\left(a \right)} + 2} - \frac{\sin^{3}{\left(a \right)} \cos{\left(a \right)} \cot{\left(a \right)}}{\cot{\left(a \right)} + 1} + \frac{\sin^{4}{\left(a \right)}}{2 \cot{\left(a \right)} + 2} + \frac{\sin^{2}{\left(a \right)} \cos^{2}{\left(a \right)}}{2 \cot{\left(a \right)} + 2} + \frac{\sin^{3}{\left(a \right)} \cos{\left(a \right)}}{\cot{\left(a \right)} + 1} + 1$$ 
1 + sin(a)^4/(2 + 2*cot(a)) + sin(a)^3*cos(a)/(1 + cot(a)) + cos(a)^2*sin(a)^2/(2 + 2*cot(a)) - sin(a)^4*cot(a)/(2 + 2*cot(a)) - sin(a)^3*cos(a)*cot(a)/(1 + cot(a)) - cos(a)^2*sin(a)^2*cot(a)/(2 + 2*cot(a))
Численный ответ [src] 
1.0 - 1.0*cos(pi/4 - a)^2*sin(a)^2*cot(pi/4 + a)
1.0 - 1.0*cos(pi/4 - a)^2*sin(a)^2*cot(pi/4 + a)
Объединение рациональных выражений [src] 
       2/pi - 4*a\    2       /pi + 4*a\
1 - cos |--------|*sin (a)*cot|--------|
        \   4    /            \   4    /
$$- \sin^{2}{\left(a \right)} \cos^{2}{\left(\frac{- 4 a + \pi}{4} \right)} \cot{\left(\frac{4 a + \pi}{4} \right)} + 1$$ 
1 - cos((pi - 4*a)/4)^2*sin(a)^2*cot((pi + 4*a)/4)
Комбинаторика [src] 
       2       2/    pi\    /    pi\
1 - sin (a)*sin |a + --|*cot|a + --|
                \    4 /    \    4 /
$$- \sin^{2}{\left(a \right)} \sin^{2}{\left(a + \frac{\pi}{4} \right)} \cot{\left(a + \frac{\pi}{4} \right)} + 1$$ 
1 - sin(a)^2*sin(a + pi/4)^2*cot(a + pi/4)
Степени [src] 
       2       2/    pi\    /    pi\
1 - sin (a)*sin |a + --|*cot|a + --|
                \    4 /    \    4 /
$$- \sin^{2}{\left(a \right)} \sin^{2}{\left(a + \frac{\pi}{4} \right)} \cot{\left(a + \frac{\pi}{4} \right)} + 1$$ 
       2/pi    \    2       /    pi\
1 - cos |-- - a|*sin (a)*cot|a + --|
        \4     /            \    4 /
$$- \sin^{2}{\left(a \right)} \cos^{2}{\left(- a + \frac{\pi}{4} \right)} \cot{\left(a + \frac{\pi}{4} \right)} + 1$$ 
                                2                              
    /   /    pi\      /     pi\\                               
    | I*|a - --|    I*|-a + --||                               
    |   \    4 /      \     4 /|                  2            
    |e             e           |  /   -I*a    I*a\     /    pi\
    |----------- + ------------| *\- e     + e   / *cot|a + --|
    \     2             2      /                       \    4 /
1 + -----------------------------------------------------------
                                 4
$$\frac{\left(e^{i a} - e^{- i a}\right)^{2} \left(\frac{e^{i \left(- a + \frac{\pi}{4}\right)}}{2} + \frac{e^{i \left(a - \frac{\pi}{4}\right)}}{2}\right)^{2} \cot{\left(a + \frac{\pi}{4} \right)}}{4} + 1$$ 
1 + (exp(i*(a - pi/4))/2 + exp(i*(-a + pi/4))/2)^2*(-exp(-i*a) + exp(i*a))^2*cot(a + pi/4)/4
Рациональный знаменатель [src] 
       2       2/    pi\    /    pi\
1 - sin (a)*sin |a + --|*cot|a + --|
                \    4 /    \    4 /
$$- \sin^{2}{\left(a \right)} \sin^{2}{\left(a + \frac{\pi}{4} \right)} \cot{\left(a + \frac{\pi}{4} \right)} + 1$$ 
       2       2/pi + 4*a\    /pi + 4*a\
1 - sin (a)*sin |--------|*cot|--------|
                \   4    /    \   4    /
$$- \sin^{2}{\left(a \right)} \sin^{2}{\left(\frac{4 a + \pi}{4} \right)} \cot{\left(\frac{4 a + \pi}{4} \right)} + 1$$ 
1 - sin(a)^2*sin((pi + 4*a)/4)^2*cot((pi + 4*a)/4)
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