Some Applications of Trigonometry

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Question

Prove the following identities where the angles involved are acute angles for which the expressions are defined.
$fraction numerator cos space straight A minus sin space straight A plus 1 over denominator cos space straight A plus sin space straight A minus 1 end fraction equals cosec space straight A plus cot space straight A.$

Solution

fraction numerator cosA minus sinA plus 1 over denominator cosA plus sinA minus 1 end fraction space equals space cosecA plus cotA

L.H.S. =

fraction numerator cosA minus sinA plus 1 over denominator cosA plus sinA minus 1 end fraction

Dividing N' and D' by sin A, we get

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

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

equals space fraction numerator cotA minus 1 plus cosecA over denominator cotA plus 1 minus cosecA end fraction equals space fraction numerator cotA plus cosecA minus 1 over denominator cotA minus cosecA plus 1 end fraction equals space fraction numerator left parenthesis cotA plus cosecA right parenthesis minus left parenthesis cosec squared straight A minus cot squared straight A right parenthesis over denominator cotA minus cosecA plus 1 end fraction equals space fraction numerator left parenthesis cotA plus cosecA right parenthesis minus left parenthesis cosecA plus cotA right parenthesis space left parenthesis cosecA minus cotA right parenthesis over denominator cotA minus cosecA plus 1 end fraction equals space fraction numerator left parenthesis cotA plus cosecA right parenthesis left parenthesis 1 minus cosecA plus cotA right parenthesis over denominator left parenthesis cotA minus cosecA plus 1 right parenthesis end fraction equals space cot space straight A space plus space cosec space straight A space equals space straight R. straight H. straight S. space Hence comma space straight L. straight H. straight S. space equals space straight R. straight H. straight S.