Some Applications of Trigonometry

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Question

Prove that:
$sec squared straight theta minus fraction numerator sin squared straight theta minus 2 sin to the power of 4 straight theta over denominator 2 space cos to the power of 4 straight theta space minus cos squared straight theta end fraction equals 1$

Solution

straight L. straight H. straight S. space equals space sec squared straight theta minus fraction numerator sin squared straight theta minus 2 sin to the power of 4 straight theta over denominator 2 cos to the power of 4 straight theta minus cos squared straight theta end fraction equals space sec squared straight theta minus fraction numerator sin squared straight theta space left parenthesis 1 minus 2 sin squared straight theta right parenthesis over denominator cos squared straight theta left parenthesis 2 cos squared straight theta minus 1 right parenthesis end fraction equals space sec squared straight theta space minus space fraction numerator sin squared straight theta open square brackets 1 minus 2 open parentheses 1 minus cos squared straight theta close parentheses close square brackets over denominator cos squared straight theta space left parenthesis 2 cos squared straight theta minus 1 right parenthesis end fraction equals space sec squared straight theta space minus tan squared straight theta fraction numerator left parenthesis 1 minus 2 plus 2 cosθ right parenthesis over denominator 2 space cosθ minus 1 end fraction space equals space sec squared straight theta space minus tan squared straight theta space open parentheses fraction numerator 2 cos squared straight theta minus 1 over denominator 2 cos squared straight theta minus 1 end fraction close parentheses equals space sec squared straight theta space minus space tan squared straight theta space equals space 1 space equals space straight R. straight H. straight S.

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Some Applications Of Trigonometry

Prove that:
$sec squared straight theta minus fraction numerator sin squared straight theta minus 2 sin to the power of 4 straight theta over denominator 2 space cos to the power of 4 straight theta space minus cos squared straight theta end fraction equals 1$

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