We simplify the given expression:
\[
\frac{\sqrt{1 + \sin x}}{1 - \sin x}
\]
Using the identity \( \sec^2 x - \tan^2 x = 1 \), we can rewrite the expression as:
\[
\frac{\sqrt{1 + \sin x}}{1 - \sin x} = \sec x \cdot \tan x
\]
Thus, the correct answer is \( \sec x \cdot \tan x \).