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Question
Prove the following trigonometric identities.
`(cos theta - sin theta + 1)/(cos theta + sin theta - 1) = cosec theta + cot theta`
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Solution
We have to prove the following identity
`(cos theta - sin theta + 1)/(cos theta + sin theta - 1) = cosec theta + cot theta`
Consider the LHS = `(cos theta - sin theta + 1)/(cos theta + sin theta - 1)`
`= (cos theta - sin theta + 1)/(cos theta + sin theta - 1) xx (cos theta + sin theta + 1)/(cos theta + sin theta + 1)`
`= ((cos theta + 1)^2 - (sin theta)^2)/((cos theta + sin theta)^2 - (1)^2)`
`= (cos^2 theta + 1 + 2 cos theta - sin^2 theta)/(cos^2 theta + sin^2 theta + 2 cos theta sin theta - 1)`
`= (cos^2 theta + 1 + 2 cos theta - (1 - cos^2 theta))/(1 + 2 cos theta sin theta - 1)`
`= (2 cos^2 theta + 2 cos theta)/(2 cos theta sin theta)`
`= (2 cos^2 theta + 2 cos theta)/(2 cos theta sin theta)`
`= (2 cos theta(cos theta + 1))/(2 cos theta sin theta)`
`= (cos theta + 1)/sin theta`
`= cos theta/sin theta + 1/sin theta`
`= cot theta + cosec theta`
= RHS
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