1. **State the problem:** Simplify the expression $$\frac{1-\sin x}{\cos x}$$. 2. **Recall the Pythagorean identity:** $$\sin^2 x + \cos^2 x = 1$$. 3. **Rewrite the numerator:** Notice that $$1 - \sin x$$ can be related to the Pythagorean identity, but here we will try to simplify by multiplying numerator and denominator by the conjugate of the numerator to rationalize it. 4. **Multiply numerator and denominator by the conjugate of the numerator:** $$\frac{1-\sin x}{\cos x} \times \frac{1+\sin x}{1+\sin x} = \frac{(1-\sin x)(1+\sin x)}{\cos x (1+\sin x)}$$. 5. **Simplify the numerator using difference of squares:** $$ (1-\sin x)(1+\sin x) = 1 - \sin^2 x $$. 6. **Use the Pythagorean identity:** $$1 - \sin^2 x = \cos^2 x$$. 7. **Substitute back:** $$\frac{\cos^2 x}{\cos x (1+\sin x)}$$. 8. **Simplify the fraction:** $$\frac{\cos^2 x}{\cos x (1+\sin x)} = \frac{\cos x}{1+\sin x}$$. **Final answer:** $$\frac{\cos x}{1+\sin x}$$.