Prove that !- \( \frac{\cos x}{1-\sin x}=\frac{1+\sin x}{\cos x} \)

To prove the equation \( \frac{\cos x}{1-\sin x}=\frac{1+\sin x}{\cos x} \), we start by cross-multiplying to eliminate the fractions: \[ \cos^2 x = (1 + \sin x)(1 - \sin x) \] The right-hand side can be rewritten using the difference of squares: \[ (1 + \sin x)(1 - \sin x) = 1^2 - (\sin x)^2 = 1 - \sin^2 x \] Using the Pythagorean identity \( 1 - \sin^2 x = \cos^2 x \), we can rewrite the equation: \[ \cos^2 x = \cos^2 x \] Since both sides are equal, the original equation is proven true. Thus: \[ \frac{\cos x}{1-\sin x} = \frac{1+\sin x}{\cos x} \]