Start with the left-hand side of the identity:

\(\left( \frac{1}{\cos x} - \tan x \right) \left( \frac{1}{\sin x} + 1 \right)\)

Substitute \(\tan x = \frac{\sin x}{\cos x}\):

\(\left( \frac{1}{\cos x} - \frac{\sin x}{\cos x} \right) \left( \frac{1}{\sin x} + 1 \right)\)

Simplify the expression:

\(\left( \frac{1 - \sin x}{\cos x} \right) \left( \frac{1 + \sin x}{\sin x} \right)\)

Multiply the numerators and denominators:

\(\frac{(1 - \sin x)(1 + \sin x)}{\cos x \sin x}\)

Use the identity \((1 - \sin x)(1 + \sin x) = 1 - \sin^2 x\):

\(\frac{1 - \sin^2 x}{\cos x \sin x}\)

Recognize \(1 - \sin^2 x = \cos^2 x\):

\(\frac{\cos^2 x}{\cos x \sin x}\)

Simplify by canceling \(\cos x\):

\(\frac{\cos x}{\sin x}\)

Recognize \(\frac{\cos x}{\sin x} = \frac{1}{\tan x}\):

Thus, the identity is proven.