TrigonometryEstablish the identity.
4 weeks ago
Establish the identity. (1 - cos theta)/(1 + cos theta) = (csc theta - cot theta)^2
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Best Answer
Christian Staff answered 4 weeks ago
Given: (1 - cos theta)/(1 + cos theta) = (csc theta - cot theta)^2 RHS(csc theta - cot theta)^2 rihgtarrow (1) csc theta = (1)/(sin theta) cot theta = (cos theta)/(sin theta) and (a -b)^2 = a^2 -2ab + b^2 from (1)(csc theta - cot theta)^2 = ((1)/(sin theta) - (cos theta)/(sin theta))^2 Apply formula ((1)/(sin theta))^2 - 2 ((1)/(sin theta))((cos theta)/(sin theta)) + ((cos theta)/(sin theta))^2 (csc theta - cot theta)^2 = (1)/(sin^2 theta) - (2 cos theta)/(sin^2 theta) + (cos^2 theta)/(sin^2 theta) -> (1) = (1 - 2 cos theta + cos^2 theta)/(sin^2 theta) = ((1 - cos theta)^2)/(sin^ 2 theta) because cos^2 theta + sin^2 theta = 1 -> sin^2 theta = 1 - cos^2 theta = ((1 - cos theta)(1 - cos theta))/((1 - cos^2 theta)) -> (2) = ((1 - cos theta)(1 - cos theta))/((1 - cos theta)(1 + cos theta)) = (1 - cos theta)/(1 + cos theta) -> (3) from (1)(2)(3) (csc theta - cot theta)^2 = (1)/(sin^2 theta) - (2cos theta)/(sin^2 theta) + (cos^2 theta)/(sin^2 theta) = ((1 - cos theta)^2)/(1 - cos^2 theta) = (1 - cos theta)/(1 + cos theta)
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