Prove that
`cos ^(4) ""(pi)/(8) + cos ^(4) ""(3pi)/(8) + cos ^(4) ""(5pi)/(8) + cos ^(4) "" (7pi)/(8) = (3)/(2)`

Answer

To prove that

\[ \cos^4\left(\frac{\pi}{8}\right) + \cos^4\left(\frac{3\pi}{8}\right) + \cos^4\left(\frac{5\pi}{8}\right) + \cos^4\left(\frac{7\pi}{8}\right) = \frac{3}{2} \]

we will follow these steps:

### Step 1: Rewrite the terms using cosine properties
We know that:
\[ \cos\left(\frac{5\pi}{8}\right) = \cos\left(\pi - \frac{3\pi}{8}\right) = -\cos\left(\frac{3\pi}{8}\right) \]
and
\[ \cos\left(\frac{7\pi}{8}\right) = \cos\left(\pi - \frac{\pi}{8}\right) = -\cos\left(\frac{\pi}{8}\right) \]

Thus, we can rewrite the expression as:
\[ \cos^4\left(\frac{\pi}{8}\right) + \cos^4\left(\frac{3\pi}{8}\right) + \left(-\cos\left(\frac{3\pi}{8}\right)\right)^4 + \left(-\cos\left(\frac{\pi}{8}\right)\right)^4 \]

This simplifies to:
\[ \cos^4\left(\frac{\pi}{8}\right) + \cos^4\left(\frac{3\pi}{8}\right) + \cos^4\left(\frac{3\pi}{8}\right) + \cos^4\left(\frac{\pi}{8}\right) \]

### Step 2: Combine like terms
Combining the terms gives:
\[ 2\cos^4\left(\frac{\pi}{8}\right) + 2\cos^4\left(\frac{3\pi}{8}\right) \]
Factoring out the 2:
\[ 2\left(\cos^4\left(\frac{\pi}{8}\right) + \cos^4\left(\frac{3\pi}{8}\right)\right) \]

### Step 3: Use the identity for cosine powers
Using the identity:
\[ \cos^4 x = \left(\cos^2 x\right)^2 = \left(\frac{1 + \cos(2x)}{2}\right)^2 \]
we can express:
\[ \cos^4\left(\frac{\pi}{8}\right) = \left(\frac{1 + \cos\left(\frac{\pi}{4}\right)}{2}\right)^2 \]
and
\[ \cos^4\left(\frac{3\pi}{8}\right) = \left(\frac{1 + \cos\left(\frac{3\pi}{4}\right)}{2}\right)^2 \]

### Step 4: Calculate values
Calculating these gives:
\[ \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}, \quad \cos\left(\frac{3\pi}{4}\right) = -\frac{1}{\sqrt{2}} \]
Thus:
\[ \cos^4\left(\frac{\pi}{8}\right) = \left(\frac{1 + \frac{1}{\sqrt{2}}}{2}\right)^2 \]
and
\[ \cos^4\left(\frac{3\pi}{8}\right) = \left(\frac{1 - \frac{1}{\sqrt{2}}}{2}\right)^2 \]

### Step 5: Substitute and simplify
Substituting these values back into the equation:
\[ 2\left(\left(\frac{1 + \frac{1}{\sqrt{2}}}{2}\right)^2 + \left(\frac{1 - \frac{1}{\sqrt{2}}}{2}\right)^2\right) \]

Calculating the squares and simplifying leads to:
\[ = 2\left(\frac{(1 + \frac{1}{\sqrt{2}})^2 + (1 - \frac{1}{\sqrt{2}})^2}{4}\right) \]

### Step 6: Final simplification
This results in:
\[ = \frac{1}{2}\left(2 + 2\cdot\frac{1}{2}\right) = \frac{3}{2} \]

### Conclusion
Thus, we have shown that:
\[ \cos^4\left(\frac{\pi}{8}\right) + \cos^4\left(\frac{3\pi}{8}\right) + \cos^4\left(\frac{5\pi}{8}\right) + \cos^4\left(\frac{7\pi}{8}\right) = \frac{3}{2} \]