Represent `cos 6 theta` in terms of `cos theta`.

Explanation

To represent \( \cos 6\theta \) in terms of \( \cos \theta \), we can use the binomial theorem and the properties of complex numbers. Here’s a step-by-step solution:

### Step 1: Use Euler's Formula
We start by expressing \( \cos 6\theta \) using Euler's formula:
\[ \cos 6\theta = \text{Re}(e^{i6\theta}) = \text{Re}((e^{i\theta})^6) = \text{Re}((\cos \theta + i \sin \theta)^6) \]

### Step 2: Expand Using the Binomial Theorem
Using the binomial theorem, we expand \( (\cos \theta + i \sin \theta)^6 \):
\[ (\cos \theta + i \sin \theta)^6 = \sum_{k=0}^{6} \binom{6}{k} (\cos \theta)^{6-k} (i \sin \theta)^k \]
This simplifies to:
\[ = \sum_{k=0}^{6} \binom{6}{k} (\cos \theta)^{6-k} (i^k)(\sin \theta)^k \]

### Step 3: Separate Real and Imaginary Parts
The real part of this expansion gives us \( \cos 6\theta \):
\[ \cos 6\theta = \sum_{k \text{ even}} \binom{6}{k} (\cos \theta)^{6-k} (i^k)(\sin \theta)^k \]
The terms with \( k \) even contribute to the real part, while those with \( k \) odd contribute to the imaginary part.

### Step 4: Calculate the Relevant Terms
Calculating the relevant terms for \( k = 0, 2, 4, 6 \):
- For \( k = 0 \):
\[ \binom{6}{0} (\cos \theta)^6 (i^0)(\sin \theta)^0 = \cos^6 \theta \]
- For \( k = 2 \):
\[ \binom{6}{2} (\cos \theta)^4 (i^2)(\sin \theta)^2 = 15 \cos^4 \theta (-1) \sin^2 \theta = -15 \cos^4 \theta \sin^2 \theta \]
- For \( k = 4 \):
\[ \binom{6}{4} (\cos \theta)^2 (i^4)(\sin \theta)^4 = 15 \cos^2 \theta (1) \sin^4 \theta = 15 \cos^2 \theta \sin^4 \theta \]
- For \( k = 6 \):
\[ \binom{6}{6} (\cos \theta)^0 (i^6)(\sin \theta)^6 = 1 \cdot (-1) \cdot \sin^6 \theta = -\sin^6 \theta \]

### Step 5: Combine the Real Parts
Now, combining these terms, we get:
\[ \cos 6\theta = \cos^6 \theta - 15 \cos^4 \theta \sin^2 \theta + 15 \cos^2 \theta \sin^4 \theta - \sin^6 \theta \]

### Step 6: Use the Identity \( \sin^2 \theta = 1 - \cos^2 \theta \)
Substituting \( \sin^2 \theta = 1 - \cos^2 \theta \):
\[ \cos 6\theta = \cos^6 \theta - 15 \cos^4 \theta (1 - \cos^2 \theta) + 15 \cos^2 \theta (1 - \cos^2 \theta)^2 - (1 - \cos^2 \theta)^3 \]

### Step 7: Simplify the Expression
Now, we simplify the expression:
1. Expand \( (1 - \cos^2 \theta)^2 \) and \( (1 - \cos^2 \theta)^3 \).
2. Combine like terms.

After simplification, we arrive at the final expression for \( \cos 6\theta \) in terms of \( \cos \theta \).

### Final Result
The final expression for \( \cos 6\theta \) in terms of \( \cos \theta \) is:
\[ \cos 6\theta = 32 \cos^6 \theta - 48 \cos^4 \theta + 18 \cos^2 \theta - 1 \]