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Some Applications Of Trigonometry

Question
CBSEENMA10008286
Wired Faculty App

Prove the following identity:
fraction numerator tanθ over denominator 1 minus cotθ end fraction plus fraction numerator cotθ over denominator 1 minus tanθ end fraction equals 1 plus tanθ plus cotθ.







Solution
straight L. straight H. straight S. space equals space fraction numerator tanθ over denominator 1 minus cotθ end fraction plus fraction numerator cot space straight theta over denominator 1 minus tanθ end fraction equals space fraction numerator begin display style fraction numerator sin space straight theta over denominator cos space straight theta end fraction end style over denominator 1 minus begin display style fraction numerator cos space straight theta over denominator sin space straight theta end fraction end style end fraction plus fraction numerator begin display style fraction numerator cos space straight theta over denominator sin space straight theta end fraction end style over denominator 1 minus begin display style fraction numerator sin space straight theta over denominator cos space straight theta end fraction end style end fraction equals space fraction numerator begin display style fraction numerator sin space straight theta over denominator cos space straight theta end fraction end style over denominator begin display style fraction numerator space sinθ minus cosθ over denominator sinθ end fraction end style end fraction plus fraction numerator begin display style cosθ over sinθ end style over denominator begin display style fraction numerator cos space straight theta space minus space sin space straight theta over denominator cos space straight theta end fraction end style end fraction equals space fraction numerator sin space straight theta over denominator cos space straight theta end fraction cross times space fraction numerator sin space straight theta over denominator sin space straight theta minus cos space straight theta end fraction plus fraction numerator cos space straight theta over denominator sin space straight theta end fraction cross times fraction numerator cos space straight theta over denominator cos space straight theta minus sin space straight theta end fraction
equals space fraction numerator sin squared straight theta over denominator cos space straight theta left parenthesis sinθ minus cosθ right parenthesis end fraction plus fraction numerator cos squared straight theta over denominator sin space straight theta space left parenthesis cosθ minus sinθ right parenthesis end fraction equals space fraction numerator sin squared straight theta over denominator cosθ left parenthesis sinθ minus cosθ right parenthesis end fraction plus fraction numerator cos squared straight theta over denominator sinθ left parenthesis cosθ minus sinθ right parenthesis end fraction equals space fraction numerator sin squared straight theta over denominator cosθ left parenthesis sinθ minus cosθ right parenthesis end fraction minus fraction numerator cos squared straight theta over denominator sinθ left parenthesis sinθ minus cosθ right parenthesis end fraction equals space fraction numerator sin cubed straight theta minus cos cubed straight theta over denominator cosθ. space sinθ space left parenthesis sinθ minus cosθ right parenthesis end fraction equals space fraction numerator left parenthesis sinθ space minus space cosθ right parenthesis thin space left parenthesis sin squared straight theta plus cos squared straight theta plus sinθ. space cosθ right parenthesis over denominator sinθ. space cosθ space left parenthesis sinθ minus cosθ right parenthesis end fraction equals space fraction numerator sin squared straight theta plus cos squared straight theta plus sinθ. space cosθ over denominator sinθ. space cosθ end fraction equals space fraction numerator sin squared straight theta over denominator sin space straight theta. space cosθ end fraction plus fraction numerator cos squared straight theta over denominator sinθ. space cosθ end fraction plus fraction numerator sinθ. space cosθ over denominator sinθ. space cosθ end fraction equals space sinθ over cosθ plus cosθ over sinθ plus 1 equals space tanθ plus cotθ plus 1 equals space straight R. straight H. straight S.
Hence, L.H.S. = R.H.S.

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