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Question
Prove the following trigonometric identities.
`(1 - sin theta)/(1 + sin theta) = (sec theta - tan theta)^2`
Solution
We have to prove `(1 - sin theta)/(1 + sin theta) = (sec theta - tan theta)^2`
We know that, `sin^2 theta + cos^2 theta = 1`
Multiplying both numerator and denominator by `(1 - sin theta)` we have
`(1 - sin theta)/(1 + sin theta) = ((1 - sin theta)(1 - sin theta))/((1 + sin theta)(1 - sin theta))`
`= (1 - sin theta)^2/(1 - sin^2 theta)`
`= ((1 - sin theta)/cos theta)^2`
`= (1/cos theta - sin theta/cos theta)^2`
`= (sec theta - tan theta)^2`
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