Question

$\frac{\cot \theta }{(cosec\theta +1)}+\frac{(cosec\theta +1)}{\cot \theta }=2\sec \theta$

Answer 1

$L.H.S=\frac{\cot \theta }{(\csc \theta +1)}+\frac{(\csc \theta +1)}{\cot \theta }$

$=\frac{\frac{\cos \theta }{\sin \theta }}{\frac{1}{\sin \theta }+1}+\frac{\frac{1}{\sin \theta }+1}{\frac{\cos \theta }{\sin \theta }}$

$=\frac{\cos \theta }{1+\sin \theta }+\frac{1+\sin \theta }{\cos \theta }$

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

$=2\sec \theta =R.H.S$ (proved).