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If cosecA+cotA=p then prove that cosA=p^2-1/p^2+1

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If cosecA+cotA=p then prove that cosA=p^2-1/p^2+1

Posted by Manoj Panth 6 years, 4 months ago

CBSE > Class 10 > Mathematics

1 answers

Sia ? 6 years, 4 months ago

Given, cosec θ + cot θ = p...(i)

We know that, {tex}cosec^2\theta-cot^2\theta=1{/tex}

{tex}\Rightarrow (cosec\theta+cot\theta)(cosec\theta-cot\theta)=1{/tex}

{tex}\Rightarrow p(cosec\theta-cot\theta)=1{/tex}

{tex}\Rightarrow cosec\theta-cot\theta=\frac 1p{/tex} ....(ii)

Adding i and ii, we get

{tex}2cosec\theta=p+ \frac 1p{/tex}

{tex}cosec\theta=\frac{p^2+1}{2p}{/tex}

{tex}\Rightarrow sin\theta= \frac{1}{cosec\theta}=\frac{2p}{p^2+1}{/tex}

We know that,{tex}cos\theta=\sqrt{1-sin^2\theta}=\sqrt{1- \frac{4p^2}{(p^2+1)^2}}=\sqrt{\frac{p^4+1-2p^2}{(p^2+1)^2}}{/tex}

{tex}cos\theta=\sqrt{\frac{(p^2-1)^2}{(p^2+1)^2}}=\frac{p^2-1}{p^2+1} {/tex}

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