Prove that:

tan 20° tan 30° tan 40° tan 80° = 1

Take L.H.S

tan 20° tan 40° tan 30° tan 80°

Multiplying & Dividing by 2:

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

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

{∵ cos (-A) = cos A}

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

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

{∵ sin (-A) = - sin A}

{∵ sin (180° - A) = sin A & cos (180° - A) = - cos A}

= 1

= R.H.S

Hence Proved