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