The problem asks us to prove the trigonometric identity: $\frac{\sin x}{1+\cos x} + \frac{1+\cos x}{\sin x} = \frac{2}{\sin x}$

1. Problem Description

The problem asks us to prove the trigonometric identity:

$\frac{\sin x}{1+\cos x} + \frac{1+\cos x}{\sin x} = \frac{2}{\sin x}$

2. Solution Steps

We will start with the left-hand side of the equation and try to simplify it to match the right-hand side.

$\frac{\sin x}{1+\cos x} + \frac{1+\cos x}{\sin x}$

Find a common denominator:

$\frac{\sin^2 x + (1+\cos x)^2}{\sin x (1+\cos x)}$

Expand the numerator:

$\frac{\sin^2 x + 1 + 2\cos x + \cos^2 x}{\sin x (1+\cos x)}$

Use the trigonometric identity $\sin^2 x + \cos^2 x = 1$:

$\frac{1 + 1 + 2\cos x}{\sin x (1+\cos x)}$

$\frac{2 + 2\cos x}{\sin x (1+\cos x)}$

Factor out a 2 from the numerator:

$\frac{2(1+\cos x)}{\sin x (1+\cos x)}$

Cancel out the common term $(1+\cos x)$:

$\frac{2}{\sin x}$

This is the same as the right-hand side of the original equation.

3. Final Answer

Therefore, we have proven that:

$\frac{\sin x}{1+\cos x} + \frac{1+\cos x}{\sin x} = \frac{2}{\sin x}$