Establish the identity. 1 - cos theta/1 + cos theta = (csc theta - cot theta)^2

Question

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

Explanation / Answer

(1- costheta)/(1+costheta) = ( csctheta - cottheta)^2

RHS : expaning the bracket : ( csctheta - cottheta)^2

= (1/sintheta - costheta/sintheta)^2

= 1/sin^2theta + cos^2theta/sin^2theta - 2costheta/sin^2theta

= (1 - costheta)^2/sin^2theta

= (1- costheta)^2/(1 - cos^2theta)

= (1-costheta)/(1+costheta)

Option D

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