The expression you've given is:

$\frac{\sqrt{\cos 2x}}{1 + \cos 2x}$

Let's simplify this step by step:

Step 1: Recognize a Trig Identity

We can use the double-angle identity for $\cos 2x$:

$\cos 2x = 2\cos^2 x - 1$

However, this identity doesn't seem to simplify the expression significantly, so we'll leave it as $\cos 2x$.

Step 2: Analyze the expression

The expression is already in its simplest form. However, it's important to check whether the value inside the square root is valid, i.e., $\cos 2x \geq 0$, to ensure that the square root exists for real numbers.

Thus, the simplified form of the expression is:

$\frac{\sqrt{\cos 2x}}{1 + \cos 2x}$

This is as simplified as it gets without further context.

Would you like more details or have any specific questions?

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Here are five related questions for further exploration:

1. What is the range of $\cos 2x$ for real values of $x$?
2. Can the expression be further simplified if $\cos 2x$ has specific values?
3. What happens to the expression when $x = \frac{\pi}{4}$?
4. How would the expression behave if $\cos 2x = 0$?
5. Could we rewrite the denominator using an alternative trigonometric identity?

Tip: Always ensure that the square root function is applied to non-negative values in real-number contexts.