`cos 2 theta ` in terms of ` cos 4 theta,`

Answer

To express \( \cos 2\theta \) in terms of \( \cos 4\theta \), we can use the double angle formula for cosine. Here’s a step-by-step solution:

### Step 1: Recall the Double Angle Formula
The double angle formula for cosine states:
\[ \cos 2a = 2\cos^2 a - 1 \]
This formula will help us express \( \cos 2\theta \) in terms of \( \cos 4\theta \).

### Step 2: Set Up the Expression
We want to express \( \cos 2\theta \) in terms of \( \cos 4\theta \). We can let \( a = 2\theta \). Therefore, we can rewrite the double angle formula as:
\[ \cos 2\theta = 2\cos^2 \theta - 1 \]
Now, we need to express \( \cos \theta \) in terms of \( \cos 4\theta \).

### Step 3: Use the Double Angle Formula Again
Using the double angle formula again, we can express \( \cos 4\theta \):
\[ \cos 4\theta = 2\cos^2 2\theta - 1 \]
Now, we can rearrange this to express \( \cos^2 2\theta \):
\[ \cos 4\theta + 1 = 2\cos^2 2\theta \]
\[ \cos^2 2\theta = \frac{\cos 4\theta + 1}{2} \]

### Step 4: Substitute Back to Find \( \cos 2\theta \)
Now we can substitute \( \cos^2 2\theta \) back into the expression for \( \cos 2\theta \):
\[ \cos 2\theta = \sqrt{\frac{\cos 4\theta + 1}{2}} \]
Since cosine can be positive or negative, we write:
\[ \cos 2\theta = \pm \sqrt{\frac{\cos 4\theta + 1}{2}} \]

### Final Expression
Thus, we have expressed \( \cos 2\theta \) in terms of \( \cos 4\theta \):
\[ \cos 2\theta = \pm \sqrt{\frac{\cos 4\theta + 1}{2}} \]