Problem Statement
Prove the trigonometric identity:
$\frac{\sin\theta}{1 + \cos\theta} + \frac{1 + \cos\theta}{\sin\theta} = 2\csc\theta$
Verified Solution & Marking Scheme
Take Common Denominator
$\text{LHS} = \frac{\sin^2\theta + (1 + \cos\theta)^2}{\sin\theta(1 + \cos\theta)}$
Expand and Use Pythagorean Identity
$\sin^2\theta + 1 + 2\cos\theta + \cos^2\theta = (\sin^2\theta + \cos^2\theta) + 1 + 2\cos\theta = 1 + 1 + 2\cos\theta = 2(1 + \cos\theta)$
Cancel Common Factor
$\frac{2(1 + \cos\theta)}{\sin\theta(1 + \cos\theta)} = \frac{2}{\sin\theta} = 2\csc\theta = \text{RHS}$