1. Problem statement: Simplify $\frac{\cos\theta}{1-\sin\theta} - \tan\theta$. 2. Rewrite the tangent using $\tan\theta = \frac{\sin\theta}{\cos\theta}$. 3. Substitute into the expression to get $\frac{\cos\theta}{1-\sin\theta} - \frac{\sin\theta}{\cos\theta}$. 4. Combine the terms over the common denominator $ (1-\sin\theta)\cos\theta $ to obtain $\frac{\cos^2\theta - \sin\theta(1-\sin\theta)}{(1-\sin\theta)\cos\theta}$. 5. Expand the numerator: $\cos^2\theta - \sin\theta(1-\sin\theta) = \cos^2\theta - \sin\theta + \sin^2\theta$. 6. Use the Pythagorean identity $\cos^2\theta + \sin^2\theta = 1$ to rewrite the numerator as $1 - \sin\theta$. 7. Cancel the common factor $1-\sin\theta$ to get $\frac{1}{\cos\theta}$. 8. Recognize $\frac{1}{\cos\theta} = \sec\theta$ and conclude the simplified result is $\sec\theta$. Final answer: $\sec\theta$.