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Some Applications Of Trigonometry

Question

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Prove the following identity:
$tan squared straight theta plus cot squared straight theta plus 2 space equals space sec squared straight theta. space cosec squared straight theta.$

Solution

tan squared straight theta plus cot squared straight theta plus 2 equals sec squared straight theta. space cosec squared straight theta

L.H.S. =

tan squared straight theta plus cot squared straight theta plus 2

equals space fraction numerator sin squared straight theta over denominator cos squared straight theta end fraction plus fraction numerator cos squared straight theta over denominator sin squared straight theta end fraction plus 2

equals space fraction numerator sin to the power of 4 straight theta plus cos to the power of 4 straight theta plus 2 sin squared straight theta. space cos squared straight theta over denominator cos squared straight theta. space sin squared straight theta end fraction equals space fraction numerator left parenthesis sin squared straight theta right parenthesis squared plus left parenthesis cos squared straight theta right parenthesis squared plus 2 sin squared straight theta. space cos squared straight theta over denominator blank end fraction equals space fraction numerator left parenthesis sin squared straight theta plus cos squared straight theta right parenthesis squared over denominator cos squared straight theta. space sin squared straight theta end fraction space equals fraction numerator 1 over denominator cos squared straight theta. space sin squared straight theta end fraction equals space sec squared straight theta. space cosec squared straight theta space equals space straight R. straight H. straight S.

Hence, L.H.S. = R.H.S.

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