Prove that this equation is true.
sin22a-sin2a=sin3a x sina
I really don't understand your question. Do you mean
$\displaystyle \sin(22a)-\sin(2a)=\sin(3a)\sin a$?
$\displaystyle \sin^2(2a)-\sin^2a=\sin^3a\sin a$?
$\displaystyle \sin^{22}a-\sin^2a=\sin^3a\sin a$?
$\displaystyle \sin^2(2a)-\sin(2a)=\sin(3a)\sin a$?
Please be more careful with your notation.
Sorry. This equation isn't true for all $\displaystyle a$, so I thought you must have meant something else. Anyway, if we let $\displaystyle a=\frac{\pi}3$, we get
$\displaystyle \sin(22a)-\sin(2a)=-\frac{\sqrt{3}}2$
but
$\displaystyle \sin(3a)\sin a = 0$.
Was there supposed to be a restriction on $\displaystyle a$?
Then you want to solve the equation for a.
My calculator comes up with 18 solutions. I would not be surprised if there were actually 22 possibles (thus leaving 4 complex solutions.) Generally an equation with $\displaystyle sin(na)$ in it will reduce to an nth degree polynomial so you would be stuck solving a 22nd degree polynomial equation, which is impossible to do in general. You are going to be stuck doing numerical approximations.
However notice that there is at least one nice solution: a = 0.
-Dan