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प्रश्न
Prove that `(cosθ)/(1 + sinθ) = (1 - sinθ)/(cosθ)`.
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उत्तर
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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Solution :
L.H.S. = cotθ + tanθ
= `cosθ/sinθ + sinθ/cosθ`
= `(square + square)/(sinθ xx cosθ)`
= `1/(sinθ xx cosθ)` ............... `square`
= `1/sinθ xx 1/square`
= cosecθ × secθ
L.H.S. = R.H.S
∴ cotθ + tanθ = cosecθ × secθ
