Some Applications of Trigonometry

Question

Prove the following identities where the angles involved are acute angles for which the expressions are defined.
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Solution

fraction numerator 1 plus secA over denominator secA end fraction equals fraction numerator sin squared straight A over denominator 1 minus cosA end fraction

L.H.S. =

fraction numerator 1 plus secA over denominator secA end fraction

equals space fraction numerator 1 plus begin display style 1 over cosA end style over denominator begin display style 1 over cosA end style end fraction space equals space fraction numerator begin display style fraction numerator cosA plus 1 over denominator cosA end fraction end style over denominator begin display style 1 over cosA end style end fraction

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R.H.S. =

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equals space fraction numerator left parenthesis 1 plus cosA right parenthesis thin space left parenthesis 1 minus cosA right parenthesis over denominator left parenthesis 1 minus cosA right parenthesis end fraction equals space 1 plus space cosA

Hence, L.H.S. = R.H.S. ALTERNATIVE METHOD:

straight R. straight H. straight S. equals fraction numerator sin squared straight A over denominator 1 minus cosA end fraction equals fraction numerator 1 minus cos squared straight A over denominator 1 minus cosA end fraction space space space space space space space space space space space space equals space fraction numerator left parenthesis 1 plus cosA right parenthesis space left parenthesis 1 minus cosA right parenthesis over denominator left parenthesis 1 minus cosA right parenthesis end fraction space space space space space space space space space space space equals space 1 plus space cosA space equals space 1 plus 1 over secA space space space space space space space space space space equals space fraction numerator secA space plus space 1 over denominator secA end fraction space equals space fraction numerator 1 plus secA over denominator secA end fraction space space space space space space space space equals straight R. straight H. straight S comma space space Hence comma space space straight L. straight H. straight S. space equals space space straight R. straight H. straight S.

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