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Expression simplification/ Least common denominator/ sin(2*x-y)*cos(y-3*x/2)-2*cot(pi/2-y)*cos(pi/2+y)*sin(y-3*pi/2)

Least common denominator sin(2*x-y)*cos(y-3*x/2)-2*cot(pi/2-y)*cos(pi/2+y)*sin(y-3*pi/2)

An expression to simplify:

The solution

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                /    3*x\        /pi    \    /pi    \    /    3*pi\
sin(2*x - y)*cos|y - ---| - 2*cot|-- - y|*cos|-- + y|*sin|y - ----|
                \     2 /        \2     /    \2     /    \     2  /
$$- \cos{\left(y + \frac{\pi}{2} \right)} 2 \cot{\left(- y + \frac{\pi}{2} \right)} \sin{\left(y - \frac{3 \pi}{2} \right)} + \sin{\left(2 x - y \right)} \cos{\left(- \frac{3 x}{2} + y \right)}$$ 
sin(2*x - y)*cos(y - 3*x/2) - (2*cot(pi/2 - y))*cos(pi/2 + y)*sin(y - 3*pi/2)
General simplification [src] 
     2         /     3*x\              
2*sin (y) + cos|-y + ---|*sin(-y + 2*x)
               \      2 /
$$2 \sin^{2}{\left(y \right)} + \sin{\left(2 x - y \right)} \cos{\left(\frac{3 x}{2} - y \right)}$$ 
2*sin(y)^2 + cos(-y + 3*x/2)*sin(-y + 2*x)
Common denominator [src] 
   /     3*x\                                       
cos|-y + ---|*sin(-y + 2*x) + 2*cos(y)*sin(y)*tan(y)
   \      2 /
$$2 \sin{\left(y \right)} \cos{\left(y \right)} \tan{\left(y \right)} + \sin{\left(2 x - y \right)} \cos{\left(\frac{3 x}{2} - y \right)}$$ 
cos(-y + 3*x/2)*sin(-y + 2*x) + 2*cos(y)*sin(y)*tan(y)
Powers [src] 
   /     3*x\                                       
cos|-y + ---|*sin(-y + 2*x) + 2*cos(y)*sin(y)*tan(y)
   \      2 /
$$2 \sin{\left(y \right)} \cos{\left(y \right)} \tan{\left(y \right)} + \sin{\left(2 x - y \right)} \cos{\left(\frac{3 x}{2} - y \right)}$$ 
    /   /    3*x\      /     3*x\\                                                                                                            
    | I*|y - ---|    I*|-y + ---||                                                                                                            
    |   \     2 /      \      2 /|                                      /   /    pi\      /     pi\\                                          
    |e              e            | /   I*(y - 2*x)    I*(-y + 2*x)\     | I*|y + --|    I*|-y - --|| /     /     3*pi\      /    3*pi\\       
  I*|------------ + -------------|*\- e            + e            /     |   \    2 /      \     2 /| |   I*|-y + ----|    I*|y - ----||       
    \     2               2      /                                      |e             e           | |     \      2  /      \     2  /|       
- ----------------------------------------------------------------- + I*|----------- + ------------|*\- e              + e            /*tan(y)
                                  2                                     \     2             2      /
$$- \frac{i \left(- e^{i \left(- 2 x + y\right)} + e^{i \left(2 x - y\right)}\right) \left(\frac{e^{i \left(- \frac{3 x}{2} + y\right)}}{2} + \frac{e^{i \left(\frac{3 x}{2} - y\right)}}{2}\right)}{2} + i \left(\frac{e^{i \left(- y - \frac{\pi}{2}\right)}}{2} + \frac{e^{i \left(y + \frac{\pi}{2}\right)}}{2}\right) \left(- e^{i \left(- y + \frac{3 \pi}{2}\right)} + e^{i \left(y - \frac{3 \pi}{2}\right)}\right) \tan{\left(y \right)}$$ 
-i*(exp(i*(y - 3*x/2))/2 + exp(i*(-y + 3*x/2))/2)*(-exp(i*(y - 2*x)) + exp(i*(-y + 2*x)))/2 + i*(exp(i*(y + pi/2))/2 + exp(i*(-y - pi/2))/2)*(-exp(i*(-y + 3*pi/2)) + exp(i*(y - 3*pi/2)))*tan(y)
Combinatorics [src] 
   /     3*x\                                       
cos|-y + ---|*sin(-y + 2*x) + 2*cos(y)*sin(y)*tan(y)
   \      2 /
$$2 \sin{\left(y \right)} \cos{\left(y \right)} \tan{\left(y \right)} + \sin{\left(2 x - y \right)} \cos{\left(\frac{3 x}{2} - y \right)}$$ 
cos(-y + 3*x/2)*sin(-y + 2*x) + 2*cos(y)*sin(y)*tan(y)
Numerical answer [src] 
cos(y - 3*x/2)*sin(2*x - y) - 2.0*cos(pi/2 + y)*cot(pi/2 - y)*sin(y - 3*pi/2)
cos(y - 3*x/2)*sin(2*x - y) - 2.0*cos(pi/2 + y)*cot(pi/2 - y)*sin(y - 3*pi/2)
Rational denominator [src] 
   /     3*x\                             /    pi\    /    pi\
cos|-y + ---|*sin(-y + 2*x) + 2*cos(y)*cos|y + --|*cot|y - --|
   \      2 /                             \    2 /    \    2 /
$$\sin{\left(2 x - y \right)} \cos{\left(\frac{3 x}{2} - y \right)} + 2 \cos{\left(y \right)} \cos{\left(y + \frac{\pi}{2} \right)} \cot{\left(y - \frac{\pi}{2} \right)}$$ 
cos(-y + 3*x/2)*sin(-y + 2*x) + 2*cos(y)*cos(y + pi/2)*cot(y - pi/2)
Combining rational expressions [src] 
   /-3*x + 2*y\                      /pi + 2*y\    /pi - 2*y\    /-3*pi + 2*y\
cos|----------|*sin(-y + 2*x) - 2*cos|--------|*cot|--------|*sin|-----------|
   \    2     /                      \   2    /    \   2    /    \     2     /
$$\sin{\left(2 x - y \right)} \cos{\left(\frac{- 3 x + 2 y}{2} \right)} - 2 \sin{\left(\frac{2 y - 3 \pi}{2} \right)} \cos{\left(\frac{2 y + \pi}{2} \right)} \cot{\left(\frac{\pi - 2 y}{2} \right)}$$ 
cos((-3*x + 2*y)/2)*sin(-y + 2*x) - 2*cos((pi + 2*y)/2)*cot((pi - 2*y)/2)*sin((-3*pi + 2*y)/2)
Assemble expression [src] 
   /    3*x\                     /pi    \    /pi    \    /    3*pi\
cos|y - ---|*sin(2*x - y) - 2*cos|-- + y|*cot|-- - y|*sin|y - ----|
   \     2 /                     \2     /    \2     /    \     2  /
$$\sin{\left(2 x - y \right)} \cos{\left(- \frac{3 x}{2} + y \right)} - 2 \sin{\left(y - \frac{3 \pi}{2} \right)} \cos{\left(y + \frac{\pi}{2} \right)} \cot{\left(- y + \frac{\pi}{2} \right)}$$ 
cos(y - 3*x/2)*sin(2*x - y) - 2*cos(pi/2 + y)*cot(pi/2 - y)*sin(y - 3*pi/2)
Expand expression [src] 
   2       /3*x\             /3*x\               2       2       /3*x\                                 2              /3*x\               2              /3*x\                                           /3*x\
sin (y)*sin|---| + cos(y)*cos|---|*sin(y) - 2*cos (x)*sin (y)*sin|---| + 2*cos(y)*sin(y)*tan(y) - 2*cos (x)*cos(y)*cos|---|*sin(y) + 2*cos (y)*cos(x)*cos|---|*sin(x) + 2*cos(x)*cos(y)*sin(x)*sin(y)*sin|---|
           \ 2 /             \ 2 /                               \ 2 /                                                \ 2 /                              \ 2 /                                           \ 2 /
$$2 \sin{\left(x \right)} \sin{\left(\frac{3 x}{2} \right)} \sin{\left(y \right)} \cos{\left(x \right)} \cos{\left(y \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(\frac{3 x}{2} \right)} \cos^{2}{\left(y \right)} - 2 \sin{\left(\frac{3 x}{2} \right)} \sin^{2}{\left(y \right)} \cos^{2}{\left(x \right)} + \sin{\left(\frac{3 x}{2} \right)} \sin^{2}{\left(y \right)} - 2 \sin{\left(y \right)} \cos^{2}{\left(x \right)} \cos{\left(\frac{3 x}{2} \right)} \cos{\left(y \right)} + \sin{\left(y \right)} \cos{\left(\frac{3 x}{2} \right)} \cos{\left(y \right)} + 2 \sin{\left(y \right)} \cos{\left(y \right)} \tan{\left(y \right)}$$ 
sin(y)^2*sin(3*x/2) + cos(y)*cos(3*x/2)*sin(y) - 2*cos(x)^2*sin(y)^2*sin(3*x/2) + 2*cos(y)*sin(y)*tan(y) - 2*cos(x)^2*cos(y)*cos(3*x/2)*sin(y) + 2*cos(y)^2*cos(x)*cos(3*x/2)*sin(x) + 2*cos(x)*cos(y)*sin(x)*sin(y)*sin(3*x/2)
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