Take L.H.S.:

Multiplying & Dividing by 2:

{∵ 2 sin A sin B = cos (A – B) – cos (A + B);

2 sin A cos B = sin (A + B) + sin (A – B) &

2 cos A cos B = cos (A + B) + cos (A – B)}

{sin (-A) = - sin A}

= tan 2A

= R.H.S.

Hence Proved