Start with the left-hand side (LHS):

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

Combine the fractions over a common denominator:

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

Expand the numerator:

\((1 + \sin x)^2 = 1 + 2\sin x + \sin^2 x.\)

Thus, the numerator becomes:

\(1 + 2\sin x + \sin^2 x + \cos^2 x.\)

Using the identity \(\sin^2 x + \cos^2 x = 1\), the numerator simplifies to:

\(2 + 2\sin x.\)

So the expression becomes:

\(\frac{2 + 2\sin x}{\cos x (1 + \sin x)}.\)

Factor out 2 from the numerator:

\(\frac{2(1 + \sin x)}{\cos x (1 + \sin x)}.\)

Cancel \(1 + \sin x\) from the numerator and denominator:

\(\frac{2}{\cos x}.\)

This matches the right-hand side (RHS), proving the identity.