Establish the identity 1sin theta1 sin theta tan theta sec

Establish the identity. 1-sin theta/1+ sin theta = (tan theta - sec theta)^2 Starting with the right side, which shows the key steps in establishing the identity? (tan theta - sec theta)^2 = tan^2 - sec^2 theta = (sin theta - 1)^2/1-sin^2 theta = 1-sin theta/1+sin theta (tan theta - sec theta)^2 = tan^2 - sec^2 theta = (sin theta - 1)^2/1-cos^2 theta = 1-sin theta/1+sin theta (tan theta - sec theta)^2 - sin^2 theta/coss^2 theta - 2 sin theta/cos^2 theta + 1/cos^2 theta = (sin theta-1)^2/1- cos^2 theta = 1 - sin theta/1 + sin theta (tan theta - sec theta)^2 - sin^2 theta/coss^2 theta - 2 sin theta/cos^2 theta + 1/cos^2 theta = (sin theta-1)^2/1- sin^2 theta = 1 - sin theta/1 + sin theta

Solution

(1-sin theta)/(1+sin theta)= (sec theta - tan theta)²

Starting with the right side

(tan theta - sec theta)²

sec theta=1/cos theta   and    tan theta= sin theta/cos theta

(sin theta/cos theta - 1/cos theta)² = (sin theta-1)²/cos²theta

                                                              = (sin theta -1)²/(1-sin²theta)= (1-sin theta)²/(1-sin theta)(1+sin theta)

                                                                                                                 =(1-sin theta)/(1+sin theta)

The correct option is D