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

1.	If cosecA+cotA=p then prove that cosA=p^2-1/p^2+1
Answer» 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}\xa0....(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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