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Question
Prove the following trigonometric identities:
(1 + tan2 θ) (1 − sin θ) (1 + sin θ) = 1
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Solution
We have to prove `(1 + tan^2 theta)(1 - sin theta)(1 + sin theta) = 1`
We know that
`sin^2 theta + cos^2 theta = 1`
`sec^2 theta - tan^2 theta = 1`
So
`(1 + tan^2 theta)(1 - sin theta) = (1 + tan^2 theta){(1 - sin theta)(1 + sin theta)}`
` = (1 + tan^2 theta)(1 - sin^2 theta)`
`= sec^2 theta cos^2 theta`
` = 1/cos^2 theta cos^2 theta`
= 1
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