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Question
Prove that `(cosθ)/(1 + sinθ) = (1 - sinθ)/(cosθ)`.
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Solution
L.H.S. = `(cosθ)/(1 + sinθ)`
= `(cosθ)/(1 + sinθ) xx (1 - sinθ)/(1 - sinθ)` ...[On rationalising the denominator]
= `(cosθ(1 - sinθ))/(1 - sin^2θ)`
= `(cosθ(1 - sinθ))/(cos^2θ)` ...`[(∵ sin^2θ + cos^2θ = 1),(∴ 1 -sin^2θ = cos^2θ)]`
= `(1 - sinθ)/(cosθ)`
= R.H.S.
∴ `(cosθ)/(1 + sinθ) = (1 - sinθ)/(cosθ)`
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