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Maharashtra State BoardSSC (English Medium) 10th Standard

Prove that cosec θ – cot θ = (sin θ)/(1 + cos θ).

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Question

Prove that cosec θ – cot θ = `(sin θ)/(1 + cos θ)`.

Theorem
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Solution

L.H.S. = cosec θ – cot θ

= `1/(sin θ) - (cos θ)/(sin θ)`

= `(1 - cos θ)/(sin θ)`

= `(1 - cos θ)/(sin θ) xx (1 + cos θ)/(1 + cos θ)`   ...[On rationalising the numerator]

= `(1 - cos^2θ)/(sinθ(1 + cosθ))`

= `(sin^2θ)/(sinθ(1 + cosθ))`   ...`[(∵ sin^2θ + cos^2θ = 1),(∴ 1 - cos^2θ = sin^2θ)]`

= `(sin θ)/(1 + cos θ)`

= R.H.S.

∴ cosec θ – cot θ = `(sin θ)/(1 + cos θ)`

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Chapter 6: Trigonometry - Q.3 (B)

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