Choose the correct option and Justify your choice.
$space fraction numerator 1 plus tan squared straight A over denominator 1 plus cot squared straight A end fraction equals$
(A) $sec squared straight A$ (B) -1 (C) $cot squared straight A$ (D) $tan squared straight A$

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Solution

fraction numerator 1 plus tan squared straight A over denominator 1 plus cot squared straight A end fraction space equals space fraction numerator sec squared straight A over denominator cosec squared straight A end fraction

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 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

equals space fraction numerator sin squared straight A over denominator cos squared straight A end fraction space equals space tan squared straight A.

Hence, correct option is (D).

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