Express `cos 6 alpha ` in terms of ` cos 3 alpha `.

Answer

To express \( \cos 6\alpha \) in terms of \( \cos 3\alpha \), we can follow these steps:

### Step 1: Rewrite \( \cos 6\alpha \)
We start by rewriting \( \cos 6\alpha \) in a form that allows us to use the double angle formula. We can express \( 6\alpha \) as \( 2 \times 3\alpha \):
\[ \cos 6\alpha = \cos(2 \times 3\alpha) \]

### Step 2: Apply the Cosine Double Angle Formula
We can use the cosine double angle formula, which states:
\[ \cos 2\theta = 2\cos^2 \theta - 1 \]
Here, we let \( \theta = 3\alpha \):
\[ \cos 6\alpha = \cos(2 \times 3\alpha) = 2\cos^2(3\alpha) - 1 \]

### Step 3: Final Expression
Thus, we can express \( \cos 6\alpha \) in terms of \( \cos 3\alpha \):
\[ \cos 6\alpha = 2\cos^2(3\alpha) - 1 \]

### Conclusion
Therefore, the expression for \( \cos 6\alpha \) in terms of \( \cos 3\alpha \) is:
\[ \cos 6\alpha = 2\cos^2(3\alpha) - 1 \]

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