Some Applications of Trigonometry

Prove the following identity: - Wired Faculty

Question

Prove the following identity:
$left parenthesis 1 plus cotA plus tanA right parenthesis space left parenthesis sinA minus cosA right parenthesis space equals space sinA. space tanA minus space cotA. space cosA.$

Solution

L.H.S. = (1+ cot A + tan A) (sin A - cos A)
$equals open parentheses 1 plus cosA over sinA plus sinA over cosA close parentheses space left parenthesis sinA minus cosA right parenthesis equals space open parentheses 1 plus fraction numerator cos squared straight A plus sin squared straight A over denominator sinA space cosA end fraction close parentheses space left parenthesis sinA minus cosA right parenthesis equals open parentheses 1 plus fraction numerator 1 over denominator sinA space cosA end fraction close parentheses space left parenthesis sinA space minus space cosA right parenthesis equals space open parentheses fraction numerator sinA. space cosA space plus 1 over denominator sinA space cosA end fraction close parentheses space left parenthesis sinA space minus space cosA right parenthesis equals space fraction numerator left parenthesis sinA minus cosA right parenthesis space left parenthesis 1 plus sinA space cosA right parenthesis over denominator sinA space cosA end fraction straight R. straight H. straight S. space equals space sinA. space tanA space minus space cotA space cosA$
$equals space sinA. space sinA over cosA minus cosA over sinA. cosA equals space fraction numerator sin squared straight A over denominator cosA end fraction minus fraction numerator cos squared straight A over denominator sinA end fraction equals fraction numerator sin cubed straight A minus cos cubed straight A over denominator sinA space cosA end fraction equals space fraction numerator left parenthesis sinA minus cosA right parenthesis space left parenthesis sin squared straight A plus cos squared straight A plus sinA. space cosA right parenthesis over denominator sinA. space cosA end fraction equals space fraction numerator left parenthesis sinA minus cosA right parenthesis left parenthesis 1 plus sinA space cosA right parenthesis over denominator sinA. cosA end fraction$
L.H.S. = R.H.S.
Hence, (1 + cot A + tan A) (sin A - cos A) = sin A . tan A - cot A . cos A Proved

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Some Applications Of Trigonometry

Prove the following identity:
$left parenthesis 1 plus cotA plus tanA right parenthesis space left parenthesis sinA minus cosA right parenthesis space equals space sinA. space tanA minus space cotA. space cosA.$

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