The value of $\sin A+\sin B=x$ and $\cos A+\cos B=y$ then $\cos \left(A–B\right)=$

$\sin A+\sin B=x$

Squaring on both sides

$\cos A+\cos B=y$

Squaring on both sides

Adding two functions

$2\cos \left(A-B\right)=x^2+y^2-2$

$\cos \left(A-B\right)=\frac{x^2+y^2-2}{2}$