Find the value of `cos^4 (pi / 8)`

Answer

To find the value of \( \cos^4 \left( \frac{\pi}{8} \right) \), we can use trigonometric identities. Here’s a step-by-step solution:

### Step 1: Use the double angle formula for cosine
We know that:
\[ \cos(2\theta) = 2\cos^2(\theta) - 1 \]
Let \( \theta = \frac{\pi}{8} \), then \( 2\theta = \frac{\pi}{4} \). Therefore:
\[ \cos\left(\frac{\pi}{4}\right) = 2\cos^2\left(\frac{\pi}{8}\right) - 1 \]

### Step 2: Substitute the known value of \( \cos\left(\frac{\pi}{4}\right) \)
We know that:
\[ \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]
Substituting this into the equation gives:
\[ \frac{1}{\sqrt{2}} = 2\cos^2\left(\frac{\pi}{8}\right) - 1 \]

### Step 3: Solve for \( \cos^2\left(\frac{\pi}{8}\right) \)
Rearranging the equation:
\[ 2\cos^2\left(\frac{\pi}{8}\right) = \frac{1}{\sqrt{2}} + 1 \]
To combine the terms on the right, we can write \( 1 \) as \( \frac{\sqrt{2}}{\sqrt{2}} \):
\[ 2\cos^2\left(\frac{\pi}{8}\right) = \frac{1 + \sqrt{2}}{\sqrt{2}} \]
Now divide both sides by 2:
\[ \cos^2\left(\frac{\pi}{8}\right) = \frac{1 + \sqrt{2}}{2\sqrt{2}} \]

### Step 4: Find \( \cos^4\left(\frac{\pi}{8}\right) \)
Now, we need to find \( \cos^4\left(\frac{\pi}{8}\right) \):
\[ \cos^4\left(\frac{\pi}{8}\right) = \left(\cos^2\left(\frac{\pi}{8}\right)\right)^2 = \left(\frac{1 + \sqrt{2}}{2\sqrt{2}}\right)^2 \]
Calculating this gives:
\[ \cos^4\left(\frac{\pi}{8}\right) = \frac{(1 + \sqrt{2})^2}{(2\sqrt{2})^2} = \frac{1 + 2\sqrt{2} + 2}{8} = \frac{3 + 2\sqrt{2}}{8} \]

### Final Answer
Thus, the value of \( \cos^4\left(\frac{\pi}{8}\right) \) is:
\[ \boxed{\frac{3 + 2\sqrt{2}}{8}} \]