Start with the left-hand side (LHS):

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

Combine the fractions:

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

Using the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\), the expression becomes:

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

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