Question

Prove that : $\frac{\csc \theta }{1+\sec \theta }+\frac{1+\sec \theta }{\csc \theta }=\frac{\sec \theta \csc \theta +\sin \theta \cos \theta +2\sin \theta }{1+\cos \theta }$

Answer 1

L.H.S$=\frac{\csc \theta }{1+\sec \theta }+\frac{1+\sec \theta }{\csc \theta }$

$=\frac{{\csc }^2\theta +{(1+\sec \theta )}^2}{\csc \theta (1+\sec \theta )}$

$=\frac{{\csc }^2\theta +1+{\sec }^2\theta +2\sec \theta }{\csc \theta (1+\sec \theta )}$

$=\frac{({\csc }^2\theta +{\sec }^2\theta )+1+2\sec \theta }{\csc \theta (1+\sec \theta )}$

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

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

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

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

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

$=\frac{\sec \theta \csc \theta +\sin \theta \cos \theta +2\sin \theta }{1+\cos \theta }=$R.H.S

Hence proved.