Prove the following identities where the angles involved are acute angles for which the expressions are defined.
$open parentheses fraction numerator 1 plus tan squared straight A over denominator 1 plus cot squared straight A end fraction close parentheses space equals space open parentheses fraction numerator 1 minus tanA over denominator 1 minus cotA end fraction close parentheses squared$

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Solution

open parentheses fraction numerator 1 plus tan squared straight A over denominator 1 plus cot squared straight A end fraction close parentheses space equals space open parentheses fraction numerator 1 minus tanA over denominator 1 minus cotA end fraction close parentheses squared

L.H.S. =

fraction numerator 1 plus tan squared straight A over denominator 1 plus cot squared straight A end fraction

space space equals space fraction numerator sec squared straight A over denominator cosec squared straight A end fraction space equals space fraction numerator begin display style fraction numerator 1 over denominator cos squared straight A end fraction end style over denominator begin display style fraction numerator 1 over denominator sin squared straight A end fraction end style end fraction space space space space equals space fraction numerator 1 over denominator cos squared straight A end fraction cross times fraction numerator sin squared straight A over denominator 1 end fraction space space space equals space fraction numerator sin squared straight A over denominator cos squared straight A end fraction space equals space tan squared straight A

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