Some Applications of Trigonometry

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Prove the following identity: - Wired Faculty

Question

Prove the following identity:
$tan squared straight A minus tan squared straight B space equals space fraction numerator sin squared straight A minus sin squared straight B over denominator cos squared straight A cross times cos squared straight B end fraction.$

Solution

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

L.H.S. =

tan squared straight A minus tan squared straight B

equals space fraction numerator sin squared straight A over denominator cos squared straight A end fraction minus fraction numerator sin squared straight B over denominator cos squared straight B end fraction equals space fraction numerator sin squared straight A. space cos squared straight B space minus space sin squared straight B. space cos squared straight A over denominator cos squared straight A. space cos squared straight B end fraction equals space fraction numerator sin squared straight A left parenthesis 1 minus sin squared straight B right parenthesis space minus space sin squared straight B left parenthesis 1 minus sin squared straight A right parenthesis over denominator cos squared straight A. space cos squared straight B end fraction

equals space fraction numerator space sin squared straight A minus sin squared straight A. space sin squared straight B space minus space sin squared straight B plus sin squared straight B. sin squared straight A over denominator cos squared straight A. space cos squared straight B end fraction equals space fraction numerator sin squared straight A minus sin squared straight B over denominator cos squared straight A. space cos squared straight B end fraction space equals space straight R. straight H. straight S.

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