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

The left-hand side (LHS) is:

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

Combine the fractions:

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

Using the identity \(s^2 + c^2 = 1\), we have \(c^2 = 1 - s^2\).

Substitute \(c^2\) in the expression:

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

Factor out \(s\):

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

Cancel \(1+s\) from numerator and denominator:

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

Thus, the LHS simplifies to \(\tan \theta\), proving the identity.