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Question
Prove the following trigonometric identities.
(cosecθ + sinθ) (cosecθ − sinθ) = cot2 θ + cos2θ
Solution
We have to prove (cosecθ + sinθ) (cosecθ − sinθ) = cot2 θ + cos2θ
We know that
`sin^2 theta + cos^2 theta = 1`
`cosec^2 theta - cot^2 theta = 1`
So,
`(cosec theta + sin theta)(cosec theta - sin theta) = cosec^2 theta - sin^2 theta`
`= (1 + cot^2 theta) - (1 - cos^2 theta)`
`= 1 + cot^2 theta - 1 + cos^2 theta`
`= cot^2 theta + cos^2 theta`
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