The problem asks us to verify the following trigonometric identity: $\frac{\cos x}{1 + \sin x} + \frac{\sin x}{1 + \cos x} = \frac{1}{2(1 + \sin x + \cos x)}$.

1. Problem Description

The problem asks us to verify the following trigonometric identity:

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

2. Solution Steps

First, let's find a common denominator for the left-hand side (LHS) of the equation:

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

Expanding the numerator, we get:

$\cos x + \cos^2 x + \sin x + \sin^2 x$

We know that $\sin^2 x + \cos^2 x = 1$. So, the numerator becomes:

$\cos x + \sin x + 1$

Expanding the denominator, we get:

$(1 + \sin x)(1 + \cos x) = 1 + \cos x + \sin x + \sin x \cos x$

Therefore, the LHS becomes:

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

Now, let's work with the right-hand side (RHS) of the equation:

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

We want to show that

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

Cross-multiplying, we have:

$2(1 + \sin x + \cos x)(1 + \sin x + \cos x) = 1 + \sin x + \cos x + \sin x \cos x$

$2(1 + \sin x + \cos x + \sin x + \sin^2 x + \sin x \cos x + \cos x + \sin x \cos x + \cos^2 x) = 1 + \sin x + \cos x + \sin x \cos x$

$2(1 + 2\sin x + 2\cos x + \sin^2 x + \cos^2 x + 2\sin x \cos x) = 1 + \sin x + \cos x + \sin x \cos x$

Since $\sin^2 x + \cos^2 x = 1$:

$2(2 + 2\sin x + 2\cos x + 2\sin x \cos x) = 1 + \sin x + \cos x + \sin x \cos x$

$4 + 4\sin x + 4\cos x + 4\sin x \cos x = 1 + \sin x + \cos x + \sin x \cos x$

$3 + 3\sin x + 3\cos x + 3\sin x \cos x = 0$

$3(1 + \sin x + \cos x + \sin x \cos x) = 0$

$1 + \sin x + \cos x + \sin x \cos x = 0$

We see a mistake in the original question. The correct equation may be

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

Or the correct equation may be $\frac{\cos x}{1+\sin x} + \frac{\sin x}{1+\cos x} = \frac{1+\sin x + \cos x}{1+\sin x+\cos x + \sin x \cos x}$.

3. Final Answer

The given trigonometric identity is incorrect.

Final Answer: The final answer is $\frac{\cos x}{1 + \sin x} + \frac{\sin x}{1 + \cos x} = \frac{1 + \sin x + \cos x}{1 + \sin x + \cos x + \sin x \cos x}$

The identity $\frac{\cos x}{1 + \sin x} + \frac{\sin x}{1 + \cos x} = \frac{1}{2(1 + \sin x + \cos x)}$ is false.