Question

please mention all the steps....​

Answer 1

Step-by-step explanation:

We have,

$y = \frac{ \sin^{2} (x) }{1 + \cos ^{2} (x) }$

$= > \frac{dy}{dx} = \frac{(1 + \cos^{2} (x)) \frac{d}{dx}( \sin^{2} (x)) - ( \sin^{2} (x) )\frac{d}{dx} (1 + \cos^{2} (x) )}{(1 + \cos^{2} (x))^{2} }$

$= > \frac{dy}{dx} = \frac{(1 + \cos^{2} (x)) \sin(2x) - ( - \sin(2x) ) \sin^{2} (x) }{ {(1 + \cos^{2} (x) )}^{2} }$

$= > \frac{dy}{dx} = \frac{(1 + \cos^{2} (x)) \sin(2x) + (\sin(2x) ) \sin^{2} (x) }{ {(1 + \cos^{2} (x) )}^{2} }$

$= > \frac{dy}{dx} = \frac{(1 + \cos^{2} (x)) \sin(2x) + (\sin(2x) )(1 - \cos^{2} (x) ) }{ {(1 + \cos^{2} (x) )}^{2} }$

$= > \frac{dy}{dx} = \frac{ \sin(2x) + \sin(2x) \cos ^{2} (x) + \sin(2x) - \sin(2x) \cos^{2} (x) }{(1 + \cos^{2} (x))^{2} }$

$= > \frac{dy}{dx} = \frac{2 \sin(2x) }{ {(1 + \cos^{2} (x)) }^{2} }$