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The expression you provided is:

\[
\cos 3\theta + \cos 5\theta + \cos 7\theta + \cos 9\theta = 4 \cos \theta \cos 2\theta \cos 6\theta
\]

### To verify this identity, we will proceed step-by-step:

#### Step 1: Using the sum of cosines formula
We can apply the sum of cosines formula for angles to simplify the left-hand side.

Using the sum formula for cosines:
\[
\cos A + \cos B = 2 \cos \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right)
\]

We first group the terms on the left-hand side:

\[
(\cos 3\theta + \cos 9\theta) + (\cos 5\theta + \cos 7\theta)
\]

Now, we apply the sum formula to each pair:

1. For \( \cos 3\theta + \cos 9\theta \):
\[
\cos 3\theta + \cos 9\theta = 2 \cos \left( \frac{3\theta + 9\theta}{2} \right) \cos \left( \frac{9\theta - 3\theta}{2} \right)
= 2 \cos(6\theta) \cos(3\theta)
\]

2. For \( \cos 5\theta + \cos 7\theta \):
\[
\cos 5\theta + \cos 7\theta = 2 \cos \left( \frac{5\theta + 7\theta}{2} \right) \cos \left( \frac{7\theta - 5\theta}{2} \right)
= 2 \cos(6\theta) \cos(\theta)
\]

#### Step 2: Combine the results
Now substitute these results back into the expression:
\[
\cos 3\theta + \cos 9\theta + \cos 5\theta + \cos 7\theta = 2 \cos(6\theta) \cos(3\theta) + 2 \cos(6\theta) \cos(\theta)
\]

Factor out \( 2 \cos(6\theta) \):
\[
= 2 \cos(6\theta) \left( \cos(3\theta) + \cos(\theta) \right)
\]

#### Step 3: Use sum of cosines again
Now apply the sum formula for \( \cos(3\theta) + \cos(\theta) \):
\[
\cos(3\theta) + \cos(\theta) = 2 \cos \left( \frac{3\theta + \theta}{2} \right) \cos \left( \frac{3\theta - \theta}{2} \right)
= 2 \cos(2\theta) \cos(\theta)
\]

#### Step 4: Final simplification
Substitute this back into the expression:
\[
2 \cos(6\theta) \left( 2 \cos(2\theta) \cos(\theta) \right)
\]
\[
= 4 \cos(6\theta) \cos(2\theta) \cos(\theta)
\]

This matches the right-hand side of the given identity:
\[
4 \cos \theta \cos 2\theta \cos 6\theta
\]

Thus, the identity is verified.

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Would you like more details or have any questions about this solution?

Here are 5 questions you might explore next:
1. Can you derive the sum of cosines formula for general angles?
2. How would this identity change if different angles were used?
3. What is the geometric interpretation of this trigonometric identity?
4. Can this identity be applied to solve real-world problems involving waves?
5. What other trigonometric identities can be verified using a similar approach?

**Tip:** Trigonometric identities can often be simplified using sum and difference formulas, so always look for opportunities to group terms effectively.

Verify that cos(3θ) + cos(5θ) + cos(7θ) + cos(9θ) = 4 cos(θ) cos(2θ) cos(6θ).

Verify the trigonometric identity cos(3θ) + cos(5θ) + cos(7θ) + cos(9θ) = 4 cos(θ) cos(2θ) cos(6θ) using the sum of cosines formula. This solution covers key trigonometric concepts and is suitable for students in Grade 11-12.

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