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if cosecA + cotA =p then prove that cosA = p×p-1/p×p+1

Posted by Anurag Sahu 5 years, 11 months ago

CBSE > Class 10 > Mathematics

1 answers

Sia ? 5 years, 11 months ago

Given, cosec θ + cot θ = p...(i)

We know that, $cosec^2\theta-cot^2\theta=1$

$\Rightarrow (cosec\theta+cot\theta)(cosec\theta-cot\theta)=1$

$\Rightarrow p(cosec\theta-cot\theta)=1$

$\Rightarrow cosec\theta-cot\theta=\frac 1p$ ....(ii)

Adding i and ii, we get

$2cosec\theta=p+ \frac 1p$

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

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

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

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

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