Question

prove that cosx/1-tanx+sinx/1-cotx=cosx+sinx

Answer 1

hope this helps you☺️

Answer 2

Hello!

L.H.S. = $\dfrac{\sf cos\:x}{\sf 1 - tan\:x} + \dfrac{\sf sin\:x}{\sf 1 - cot\:x}$

= $\dfrac{\sf cos^{2}x}{\sf cos\:x(1 - tan\:x)} + \dfrac{\sf sin^{2}x}{\sf sin\:x(1 - cot\: x)}$

= $\dfrac{\sf cos^{2}x}{\sf cos\:x - sin\:x} + \dfrac{\sf sin^{2}x}{\sf sin\:x - cos\:x}$

= $\dfrac{\sf cos^{2}x}{\sf cos\:x - sin\:x} - \dfrac{\sf sin^{2}x}{\sf cos\:x - sin\:x}$

= $\dfrac{\sf cos^{2}x - sin^{2}x}{\sf cos\:x - sin\:x}$

= $\dfrac{\sf (cos\:x + sin\:x) \cancel{(cos\:x - sin\:x)}}{\sf \cancel{cos\:x - sin\:x}}$

= $\sf cos\:x + sin\:x$ = R.H.S.

$\boxed{\sf HENCE\: PROVED}$

Cheers!