Question

Prove that $\frac{\sin \theta }{1-\cot \theta }+\frac{\cos \theta }{1-\tan \theta }=\cos \theta +\sin \theta$.

Answer 1

Given : $\frac{\sin \theta }{1-\cot \theta }+\frac{\cos \theta }{1-\tan \theta }=\cos \theta +\sin \theta$
L.H.S. $=\frac{\sin \theta }{1-\frac{\cos \theta }{\sin \theta }}+\frac{\cos \theta }{1-\frac{\sin \theta }{\cos \theta }}$
$=\frac{{\sin }^2\theta }{\sin \theta -\cos \theta }+\frac{{\cos }^2\theta }{\cos \theta -\sin \theta }$
$=\frac{{\sin }^2\theta -{\cos }^2\theta }{\sin \theta -\cos \theta }$
$=\frac{(\sin \theta +\cos \theta )(\sin \theta -\cos \theta )}{\sin \theta -\cos \theta }$
$=\sin \theta +\cos \theta =$ R.H.S.
L.H.S $=$ R.H.S.