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प्रश्न
Prove that `costheta/(1 + sintheta) = (1 - sintheta)/(costheta)`
उत्तर
L.H.S = `costheta/(1 + sintheta)`
= `costheta/(1 + sintheta) xx (1 - sintheta)/(1 - sintheta)` ......[On rationalising the denominator]
= `(costheta(1 - sintheta))/(1 - sin^2theta)`
= `(costheta(1 - sintheta))/(cos^2theta)` ......`[(because sin^2theta +cos^2theta = 1),(therefore 1 -sin^2theta = cos^2theta)]`
= `(1 - sintheta)/costheta`
= R.H.S
∴ `costheta/(1 + sintheta) = (1 - sintheta)/(costheta)`
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L.H.S = `square`
= `square/sintheta + sintheta/costheta`
= `(cos^2theta + sin^2theta)/square`
= `1/(sintheta*costheta)` ......`[cos^2theta + sin^2theta = square]`
= `1/sintheta xx 1/square`
= `square`
= R.H.S
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