Find $\cos(x+y)$ and $\sin(x+y)$ given that $\cos x + \cos y = a$ and $\sin x + \sin y = b$ If $\cos x + \cos y = a$ and $\sin x + \sin y = b$.
Find $\cos(x+y)$  and $\sin(x+y)$. 
I only need some hints to start as I am not able to get any way to go forward to.

HINT:
Let $u = \cos x + i \sin x$, $v = \cos y + i \sin y$. Then 
\begin{eqnarray}
u + v = a+ i b \\
\frac{1}{u} + \frac{1}{v} = a -i b
\end{eqnarray}
so 
$$u v = (a+ib)/(a-ib)$$
But $uv = \cos(x+y) + i \sin(x+y)$. Now take real and imaginary parts.
