Question

cosA/1 - tanA + sinA/ 1- cotA = sinA + cosA
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Answer 1

SOLUTION :-

$\\ \\ \sf \: l.h.s = \frac{cosA}{1 - tanA} + \frac{sinA}{1 - cotA} \\ \\ \\ \bigstar \boxed{ \bf \:tanA = \frac{sinA}{cosA} } \: \: \: \: \: \: \: \bigstar\boxed{ \bf \: cotA = \frac{cosA}{sinA} } \\ \\ \\ \implies \sf \: \frac{cosA}{1 - \frac{sinA}{cosA} } + \frac{sinA}{1 - \frac{cosA}{sinA} } \\ \\ \\ \implies \sf \: \frac{cosA}{ \frac{cosA - sinA}{cosA} } + \frac{sinA}{ \frac{sinA - cosA}{sinA} } \\ \\ \\ \implies \sf \: \frac{ {cos}^{2}A }{cosA - sinA} + \frac{ {sin}^{2}A }{sinA - cosA} \\ \\ \\ \implies \sf \: \frac{ {cos}^{2}A }{cosA - sinA} - \frac{ {sin}^{2}A }{cosA - sinA} \\ \\ \\ \implies \sf \: \frac{ {cos}^{2}A - {sin}^{2}A }{cosA - sinA} \\ \\ \\ \implies \sf \: \frac{ \cancel{(cosA - sinA)}(cosA + sinA)}{ \cancel{cosA - sinA} } \\ \\ \\ \implies\sf \: cosA \: + \: sinA \: = r.h.s \: \: \: \: \: \: {(verified)}$