Simplify $$\frac{\text{cos}\space x}{1-\text{sin}\space x}$$​ using trigonometric identities:

Correct option is A

​$$\begin{aligned}&\text{Multiply both the numerator and the denominator by the conjugate of the denominator, which is } 1 + \sin x: \\&\qquad \frac{\cos x}{1 - \sin x} \cdot \frac{1 + \sin x}{1 + \sin x} = \frac{\cos x (1 + \sin x)}{(1 - \sin x)(1 + \sin x)} \\\\&\textbf{Simplify the Denominator} \\\\&\text{The denominator is a difference of squares:} \\&\qquad (1 - \sin x)(1 + \sin x) = 1 - \sin^2 x \\\\&\text{Using the Pythagorean identity } 1 - \sin^2 x = \cos^2 x: \\&\qquad \frac{\cos x (1 + \sin x)}{\cos^2 x} \\\\&\textbf{Cancel } \cos x \text{ in the Numerator and Denominator} \\&\qquad \frac{1 + \sin x}{\cos x} \\\\&\text{This can be written as:} \\&\qquad \frac{1}{\cos x} + \frac{\sin x}{\cos x} = \sec x + \tan x\end{aligned}$$​​