Problem Statement
Prove that: $\frac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \frac{1}{\sec\theta - \tan\theta}$.
Verified Solution & Marking Scheme
Divide by cos θ
$\text{LHS} = \frac{\tan\theta + \sec\theta - 1}{\tan\theta - \sec\theta + 1}$
Substitute 1 = sec² θ - tan² θ
$\text{Num} = (\sec\theta + \tan\theta) - (\sec^2\theta - \tan^2\theta) = (\sec\theta + \tan\theta)(1 - \sec\theta + \tan\theta)$
Cancel Denominator
$\text{LHS} = \sec\theta + \tan\theta = \frac{\sec^2\theta - \tan^2\theta}{\sec\theta - \tan\theta} = \frac{1}{\sec\theta - \tan\theta}$