Prove that ()/(1-) = + .

Here are the solutions to the problems from the image.

9. PURE MATHEMATICS

a) Prove that $\frac{\cos\theta}{1-\sin\theta} = \sec\theta + \tan\theta$.

Step 1: Start with the left-hand side (LHS) and multiply the numerator and denominator by the conjugate of the denominator.

$$\text{LHS} = \frac{\cos\theta}{1-\sin\theta}$$ $$\text{LHS} = \frac{\cos\theta}{1-\sin\theta} \times \frac{1+\sin\theta}{1+\sin\theta}$$

Step 2: Expand the expression.

$$\text{LHS} = \frac{\cos\theta(1+\sin\theta)}{1^2 - \sin^2\theta}$$

Recall the identity $\cos^2\theta + \sin^2\theta = 1$, so $1 - \sin^2\theta = \cos^2\theta$.

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