Answer
See below.
Work Step by Step
Using the given identities: $\frac{1+\cos x}{\sin x }+\frac{\sin x}{1+\cos x}=\frac{(1+\cos x)^2+\sin^2 x}{\sin x(1+\cos x)}=\frac{1+2\cos x+\cos^2 x+\sin^2 x}{\sin x(1+\cos x)}=\frac{1+2\cos x+1}{\sin x(1+\cos x)}=\frac{2+2\cos x}{\sin x(1+\cos x)}=\frac{2(1+\cos x)}{\sin x(1+\cos x)}=\frac{2}{\sin x}=2\csc x$
Hence we showed what we had to.
