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Sec A + tanA =P, find the value of cosec A?

Answer» {tex}sec\\ \\theta+ tan\\ \\theta = p{/tex} ...(i)Also {tex}sec^2\xa0\\theta - tan^2 \\theta = 1{/tex}{tex}\\Rightarrow{/tex}\xa0(sec\xa0{tex}\\theta{/tex}\xa0- tan\xa0{tex}\\theta{/tex}) (sec\xa0{tex}\\theta{/tex}\xa0+ tan\xa0{tex}\\theta{/tex}) = 1{tex}\\Rightarrow{/tex}\xa0p(sec\xa0{tex}\\theta{/tex}\xa0-\xa0tan\xa0{tex}\\theta{/tex}) = 1[using equation (i)]{tex}\\Rightarrow{/tex}\xa0sec\xa0{tex}\\theta{/tex}\xa0-\xa0tan\xa0{tex}\\theta{/tex}\xa0{tex}=\\frac{1}{p}{/tex}\xa0...(ii)(ii) - (i) we get{tex}-2 tan{/tex}\xa0{tex}\\theta{/tex}\xa0{tex}=\\frac{1-p^{2}}{p}{/tex}{tex}\\Rightarrow{/tex}- tan\xa0{tex}\\theta{/tex}\xa0{tex}=\\frac{1-p^{2}}{2 p}{/tex}{tex}\\Rightarrow{/tex}- cot\xa0{tex}\\theta{/tex}\xa0{tex}=\\frac{2 p}{1-p^{2}}{/tex}cot\xa0{tex}\\theta{/tex}\xa0{tex}=\\left(\\frac{2 p}{1-p^{2}}\\right)^{2}{/tex}{tex}=\\frac{-4 p^{2}}{\\left(1-p^{2}\\right)^{2}}{/tex}{tex}\\Rightarrow{/tex}\xa0{tex}cosec^2{/tex}\xa0{tex}\\theta{/tex}\xa0- 1\xa0{tex}=\\frac{-4 p^{2}}{\\left(1-p^{2}\\right)^{2}}{/tex}{tex}\\Rightarrow{/tex}\xa0{tex}cosec^2{/tex}\xa0{tex}\\theta{/tex}\xa0{tex}=\\frac{-4 p^{2}}{\\left(1-p^{2}\\right)^{2}}+1=\\frac{-4 p^{2}+\\left(1-p^{2}\\right)^{2}}{\\left(1-p^{2}\\right)^{2}}{/tex}{tex}cosec^2{/tex}\xa0{tex}\\theta{/tex}\xa0{tex}=\\frac{-4 p^{2}+1+p^{4}-2 p^{2}}{\\left(1-p^{2}\\right)^{2}}{/tex}{tex}=\\frac{\\left(1+p^{2}\\right)^{2}}{\\left(1-p^{2}\\right)^{2}}{/tex}{tex}\\Rightarrow{/tex}\xa0{tex}cosec\\\xa0\\theta{/tex}\xa0{tex}=\\frac{1+p^{2}}{1-p^{2}}{/tex}


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