Question

cotA+cosecA-1/cotA-cosecA+1=1+cosA/sinA

Answer 1

SOLUTION:-

Take L.H.S

$\frac{cot \: A + cosec \: A - 1}{cot \: A - cosec \: A + 1} \\ \\ = > \frac{cot \: A + cosec \: A - ( {cosec}^{2} A - {cot}^{2} A)}{cot \: A - cosec \: A + 1} \: \: \: \: \: \: [ {cosec}^{2} \theta - {cot}^{2} \theta = 1]\\ \\ = > \frac{(cot \: A + cosec \: A) - (cosec \: A - cot \: A)(cosec \: A + cot \: A)}{cot \: A - cosec \: A + 1} \\ \\ = > \frac{(cosec \: A + cot \: A)[(1 - (cot \: A - cosec \: A)]}{cot \: A - cosec \: A + 1} \\ \\ = > \frac{(cosec \: a + cot \: a)[cot \: A - cosec \: A + 1]}{cot \: A - cosec \: A + 1} \\ \\ = > cosec \: A + cot \: A \\ \\ = > \frac{1}{sin \: a} + \frac{cos \: A}{sin \: A} \\ \\ = > \frac{1 + co s \: A}{sin \: A} \: \: \: \: \: \: [R.H.S]$

Hope it helps ☺️