If sin m + sin n = p, cos m + cos n = q, then find the value of sin m × sin n + cos m ×cos n.

$p^2 + q^2 - 2/2$

sin m + sin n = p

On squaring both side,

( sin m + sin n )² = p² 

sin² m + sin² n + 2 × sin m × sin n = P²   ----(1)

And  cos m + cos n = q

On squaring both side,

( cos m + cos n )² = q² 

cos² m + cos² n + 2 × cos m × cos n = q²   ----(2)

On adding equation 1 and 2.

sin² m + sin² n + 2 × sin m × sin n + cos² m + cos² n + 2 × cos m × cos n =  P² + q² 
2 + 2 × sin m × sin n + 2 × cos m × cos n = P² + q²

sin m × sin n +  cos m × cos n = \(\frac{P² + q² - 2  }{2}\)