Start with the left-hand side:

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

Combine over a common denominator:

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

Expand the numerators:

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

Simplify the numerator:

\(\frac{4 \\sin x}{1 - \\sin^2 x}\)

Since \(1 - \\sin^2 x = \\cos^2 x\), the expression becomes:

\(\frac{4 \\sin x}{\\cos^2 x}\)

Rewrite \(\\sin x / \\cos x\) as \(\\tan x\):

\(\frac{4 \\tan x}{\\cos x}\)

This matches the right-hand side, proving the identity.