Question

Differentiate the function w.r.t.x:
$\rm \frac{1}{cosec\ x+\cot x}$

Answer 1

HOPE IT HELPS U ✌️✌️

Answer 2

We know that such form of expression can solve by U/V method of differentiation

$\frac{d}{dx} \bigg(\frac{U}{V}\bigg ) = \frac{V \frac{dU}{dx} - U \frac{dV}{dx} }{ {V}^{2} } \\ \\ here \: U = 1 \\ \\ V = cosec\:x+cot \: x$
$\frac{d}{dx} \bigg(\frac{1}{cosec\:x+cot \: x} \bigg) = \frac{(cosec\:x+cot \: x) \frac{d(1)}{dx} -( 1 )\frac{d(cosec\:x+cot \: x)}{dx} }{( {cosec\:x+cot \: x})^{2} }$

$\frac{d}{dx} \bigg(\frac{1}{cosec\:x+cot \: x} \bigg) = \frac{(cosec\:x+cot \: x)(0) -( 1 )(-cosec\:x\:cot\:x-{cosec}^{2} x) }{( {cosec\:x+cot \: x})^{2} } \\ \\ \frac{d}{dx} \bigg(\frac{1}{cosec\:x+cot \: x} \bigg)= \frac{cosec\:x(cosec\:x+cot\:x)}{( {cosec\:x+cot \: x})^{2} } \\ \\ \frac{d}{dx} \bigg(\frac{1}{cosec\:x+cot \: x} \bigg)= \frac{cosec\:x}{( {cosec\:x+cot \: x})}$

Hope it helps you.