Question

Find A and B if sin2xcos3x = A sin 5x + B sin x

Answer 1

Recall that

$\sin(x+y)=\sin x\cos y+\cos x\sin y$
$\sin(x-y)=\sin x\cos y-\cos x\sin y$

Adding these together, you have

$\sin(x+y)+\sin(x-y)=2\sin x\cos y$
$\dfrac12\sin(x+y)+\dfrac12\sin(x-y)=\sin x\cos y$

Replace $x$ with $2x$ and $y$ with $3x$. You end up with

$\dfrac12\sin(2x+3x)+\dfrac12\sin(2x-3x)=\sin2x\cos3x$
$\sin2x\cos3x=\dfrac12\sin5x+\dfrac12\sin(-x)=\dfrac12\sin5x-\dfrac12\sin x$

and so $A=\dfrac12$ and $B=-\dfrac12$.