Prove the identity tan theta 1 cos 2 thetasin 2 thetaSolut

Prove the identity: tan theta = 1 - cos (2 theta)/sin (2 theta)

Solution

tan theta=(1-cos 2theta)/sin 2theta

Starting with left side

tan theta

using half angle identity

sin theta/cos theta

sqrt(1-cos2theta)/sqrt(1+cos 2 theta)

sqrt(1-cos 2theta)(1-cos 2 theta)/(1-cos²2theta)

sqrt((1-cos2theta)²/sin²2theta)= (1-cos2theta)/sin2theta