Let \(s = \sin x\) and \(c = \cos x\).

Start with the left-hand side:

\(\frac{s}{1 - s} - \frac{s}{1 + s}\)

Combine the fractions:

\(\frac{s(1 + s) - s(1 - s)}{(1 - s)(1 + s)} = \frac{s + s^2 - s + s^2}{1 - s^2} = \frac{2s^2}{1 - s^2}\)

Use the identity \(1 - s^2 = c^2\):

\(\frac{2s^2}{c^2}\)

Since \(\tan x = \frac{s}{c}\), we have:

\(\frac{2s^2}{c^2} = 2 \left( \frac{s}{c} \right)^2 = 2 \tan^2 x\)

Thus, the identity is proven.