Sir please tell me some easy method to prove the following:-

cos (x + y)
=sin (90 - (x + y))
=sin (90 - x - y)
=sin (90 - x) cos (-y) + sin (-y) cos (90 - x)
=cos x cos y - sin x sin y ...(i)

substituting ‘y’ by ‘-y’ in (i)

cos (x-y) = cos x cos (-y) - sin x sin (-y)

            = cos x cos y + sin x sin y  {since  cos(-x)=cos x and sin (-x) = -sin x}

 

sin (x + y)
=cos (90 - (x + y))
=cos ((90 - x ) - y)
=cos (90 - x) cos y + sin y sin (90 - x)
=sin x cos y + cos x sin y ...(i)

substituting ‘y’ by ‘-y’ in (i)

sin (x-y) = sin x cos (-y) + cos x sin (-y)

            = sin x cos y - cos x sin y  {since  cos(-x)=cos x and sin (-x) = -sin x}