Answer
See below.
Work Step by Step
Using the learned identities: $(1+\sec x)(1-\cos x)=(1+\frac{1}{\cos x})(1-\cos x)=1-\cos x+\frac{1}{\cos x}-1=\frac{1}{\cos x}-\cos x=\frac{1}{\cos x}-\frac{\cos^2 x}{\cos x}=\frac{1-\cos^2 x}{\cos x}=\frac{\sin^2 x}{\cos x}=\tan^2 x\cos x$
Hence we proved what we had to.