证明 $\frac{1 + \cos \theta}{\sin \theta} + \frac{\sin \theta}{1 + \cos \theta} = 2 \csc \theta$。

1. Combine the fractions on the left-hand side over a common denominator.

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

2. Expand the numerator.

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

3. Apply the Pythagorean identity to simplify the numerator.

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

4. Factorise the numerator and simplify to reach the right-hand side.

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