<s>[INST] 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.
 [/INST] Using Prosthaphaeresis Formulas
$$2\sin\frac{x+y}2\cos\frac{x-y}2=a$$
and $$2\cos\frac{x+y}2\cos\frac{x-y}2=b$$
Divide to find $\tan\dfrac{x+y}2$ assuming $ab\cos\dfrac{x-y}2\ne0$ 
Now apply Weierstrass substitution

Alternatively find $a^2+b^2,a^2-b^2,ab$ and use Prosthaphaeresis Formulas
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