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Some Applications Of Trigonometry

Question

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Prove the following identity:
$square root of fraction numerator 1 plus cosθ over denominator 1 minus cosθ end fraction end root equals space cosecθ plus cotθ.$

Solution

straight L. straight H. straight S. space equals space square root of fraction numerator 1 plus cosθ over denominator 1 minus cosθ end fraction end root space space space space space space space space space space space space space space equals square root of fraction numerator 1 plus cosθ over denominator 1 minus cosθ end fraction cross times fraction numerator 1 plus cosθ over denominator 1 plus cosθ end fraction end root space space space space space space space space space space space space space equals space square root of fraction numerator left parenthesis 1 plus cosθ right parenthesis squared over denominator 1 minus cos squared straight theta end fraction end root space equals space square root of fraction numerator left parenthesis 1 plus cosθ right parenthesis squared over denominator sin squared straight theta end fraction end root space space space space space space space space space space space space space equals space square root of open parentheses fraction numerator 1 plus cosθ over denominator sinθ end fraction close parentheses squared end root space equals space open parentheses fraction numerator 1 plus cosθ over denominator sin space straight theta end fraction close parentheses space space space space space space space space space space space space space space equals space 1 over sinθ plus cosθ over sinθ space equals space cosecθ plus cotθ space space space space space space space space space space space space space space equals space straight R. straight H. straight S.

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