Let \(s = \sin x\) and \(c = \cos x\). Then \(\tan x = \frac{s}{c}\).

Substitute \(\tan x = \frac{s}{c}\) into the left-hand side:

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

Simplify the expression:

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

Since \(s^2 + c^2 = 1\), the expression becomes:

\(= s + c\)

This matches the right-hand side, \(\sin x + \cos x\).