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