Some Applications of Trigonometry

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Question

If $fraction numerator tan space straight A over denominator tan space straight B end fraction equals straight n space and space fraction numerator sin space straight A over denominator sin space straight B end fraction equals straight m comma$ then show that $cos squared straight A space equals space fraction numerator straight m squared minus 1 over denominator straight n squared minus 1 end fraction.$

Solution

space space space space space space space space space space space space space tanA space equals space straight n space tan space straight B rightwards double arrow space space space space space space space space tan space straight B space equals space 1 over straight n tanA rightwards double arrow space space space space space space space space cot space straight B space equals space fraction numerator straight n over denominator tan space straight A end fraction

and space space space space sin space straight A space equals space straight m space sinB rightwards double arrow space space space space space space sin space straight B space equals space 1 over straight m sinA rightwards double arrow space space space space space space cosecB space equals space fraction numerator straight m over denominator sin space straight A end fraction

Substituting the values of cot B and cosec B in cosec² B - cot² B = 1, we get

rightwards double arrow space space space space space space space space space fraction numerator straight m squared over denominator sin squared straight A end fraction minus fraction numerator straight n squared over denominator tan squared straight A end fraction equals 1 rightwards double arrow space space space space space fraction numerator straight m squared over denominator sin squared straight A end fraction minus fraction numerator straight n squared space cos squared straight A over denominator sin squared straight A end fraction equals 1 rightwards double arrow space space space space space space space fraction numerator straight m squared minus straight n squared space cos squared straight A over denominator sin squared straight A end fraction equals 1 rightwards double arrow space space space space space space space space space space space straight m squared minus straight n squared cos squared straight A space equals space sin squared straight A rightwards double arrow space space space space space space space space space space space space straight m squared minus straight n squared cos squared straight A space equals space 1 minus cos squared straight A rightwards double arrow space space space space space space space space space space space space straight m squared minus 1 space equals space straight n squared cos squared straight A space minus space cos squared straight A rightwards double arrow space space space space space space space space space space space space fraction numerator straight m squared minus 1 over denominator straight n squared minus 1 end fraction equals cos squared straight A

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