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?

---

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.

Explore the step-by-step simplification of √(cos 2x) / (1 + cos 2x) using trigonometric identities such as the double angle identity for cos(2x). This problem focuses on trigonometric concepts and simplification, ideal for Grades 11-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation