Question

If $\cos x+\cos y=a$ and $\sin x+\sin y=b$ then find the value of $\cos (x-y)$.

Answer 1

Given,$\cos x+\cos y=a$.....(1) and $\sin x+\sin y=b$......(2).

Now squaring (1) and (2) and then adding we get,

$(\cos x+\cos y{)}^2+(\sin x+\sin y{)}^2=a^2+b^2$

or, $({\sin }^{2}x+{\cos }^{2}x)+({\sin }^{2}y+{\cos }^{2}y)+2(\cos x.\cos y+\sin x.\sin y)=a^2+b^2$

or, $2\cos (x-y)=(a^2+b^2-2)$

or, $\cos (x-y)=\frac{(a^2+b^2-2)}{2}$