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The image you uploaded shows a trigonometric identity verification problem, and it appears to be guiding the user through the process of simplifying the left-hand side of the identity using trigonometric identities, particularly the double-angle identities. Let's work through this step by step.

### The given identity:
\[
\frac{3 + 3\cos(2x)}{\sin(2x)} = 3 \cot(x)
\]

We need to simplify the left-hand side to show that it equals the right-hand side.

### Step-by-step solution:

1. **Left-hand side**:
   \[
   \frac{3 + 3\cos(2x)}{\sin(2x)}
   \]

2. **Factor out 3**:
   \[
   = 3 \frac{1 + \cos(2x)}{\sin(2x)}
   \]

3. **Use the double-angle identity for \(\cos(2x)\)**:
   \[
   \cos(2x) = 1 - 2\sin^2(x)
   \]
   Therefore,
   \[
   1 + \cos(2x) = 1 + (1 - 2\sin^2(x)) = 2 - 2\sin^2(x) = 2\cos^2(x)
   \]

   So, the expression becomes:
   \[
   = 3 \frac{2\cos^2(x)}{\sin(2x)}
   \]

4. **Use the double-angle identity for \(\sin(2x)\)**:
   \[
   \sin(2x) = 2\sin(x)\cos(x)
   \]

   Substituting this into the denominator:
   \[
   = 3 \frac{2\cos^2(x)}{2\sin(x)\cos(x)}
   \]

5. **Simplify**:
   The \(2\)'s cancel out, and one \(\cos(x)\) cancels from the numerator and denominator:
   \[
   = 3 \frac{\cos(x)}{\sin(x)}
   \]

6. **Final simplification**:
   \[
   = 3 \cot(x)
   \]

Thus, the left-hand side simplifies to the right-hand side, verifying the identity.

### Completed steps:

\[
\frac{3 + 3\cos(2x)}{\sin(2x)} = 3\cot(x)
\]

---

Do you want more details or have any questions about this process?

Here are 5 relative questions for practice:
1. Can you prove the identity \(\sin(2x) = 2\sin(x)\cos(x)\) from first principles?
2. What is the double-angle identity for \(\tan(2x)\)?
3. How would the problem change if \(\sin(x)\) instead of \(\cos(x)\) was in the original equation?
4. Can you derive other trigonometric identities from \(\cos(2x) = 1 - 2\sin^2(x)\)?
5. How can you express \(\cot(x)\) in terms of \(\sin(x)\) and \(\cos(x)\)?

**Tip:** Double-angle identities are often used in trigonometric simplifications and can be crucial for solving complex problems involving sine and cosine functions.

Verify the identity (3 + 3cos(2x))/sin(2x) = 3cot(x)

Learn how to verify the trigonometric identity (3 + 3cos(2x))/sin(2x) = 3cot(x) using double-angle identities. This step-by-step solution demonstrates key trigonometric concepts, including simplifications and double-angle formulas. Suitable for Grades 10-11 students.

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