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प्रश्न
Prove the following identities, where the angles involved are acute angles for which the expressions are defined:
`sqrt((1+sinA)/(1-sinA)) = secA + tanA`
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उत्तर
L.H.S
= `sqrt((1+sinA)/(1-sinA))`
= `sqrt(((1+sinA)(1+sinA))/((1-sinA)(1+sinA))`
= `(1+sinA)/(sqrt(1-sin^2A))`
= `(1+sinA)/sqrt(cos^2A)`
= `(1+sinA)/cosA`
= secA + tan A
= `1/cos A + sin A/cos A`
= R.H.S
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Activity:
L.H.S = `square`
= `cos^2theta xx square .....[1 + tan^2theta = square]`
= `(cos theta xx square)^2`
= 12
= 1
= R.H.S
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Activity: L.H.S. = cotθ + tanθ
= `cosθ/sinθ + square/cosθ`
= `(square + sin^2theta)/(sinθ xx cosθ)`
= `1/(sinθ xx cosθ)` ....... ∵ `square`
= `1/sinθ xx 1/cosθ`
= `square xx secθ`
∴ L.H.S. = R.H.S.
