Prove the following identity:
$fraction numerator sinA minus sinB over denominator cosA plus cosB end fraction plus fraction numerator cosA minus cosB over denominator sinA plus sinB end fraction equals 0.$

Solution

L.H.S. = $fraction numerator sinA minus sinB over denominator cosA plus cosB end fraction plus fraction numerator cosA plus cosB over denominator sinA plus sinB end fraction$
= $fraction numerator left parenthesis sin squared straight A minus sin squared straight B right parenthesis plus left parenthesis cos squared straight A minus cos squared straight B right parenthesis over denominator left parenthesis cosA plus cosB right parenthesis thin space left parenthesis sinA plus sinB right parenthesis end fraction$
$equals space fraction numerator left parenthesis sin squared straight A plus cos squared straight A right parenthesis space minus space left parenthesis sin squared straight B plus cos squared straight B right parenthesis over denominator left parenthesis cosA plus cosB right parenthesis thin space left parenthesis sinA plus sinB right parenthesis end fraction equals space fraction numerator 1 minus 1 over denominator left parenthesis cosA plus cosB right parenthesis space left parenthesis sinA plus sinB right parenthesis end fraction equals space fraction numerator 0 over denominator left parenthesis cosA plus cosB right parenthesis thin space left parenthesis sinA plus sinB right parenthesis end fraction equals space 0. space equals space straight R. straight H. straight S.$
Hence, L.H.S. = R.H.S.

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Some Applications Of Trigonometry

Prove the following identity:
$fraction numerator sinA minus sinB over denominator cosA plus cosB end fraction plus fraction numerator cosA minus cosB over denominator sinA plus sinB end fraction equals 0.$

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