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

Options

A. `(1)/(2)`
B. `(2)/(2)`
C. `(3)/(2)`
D. `(4)/(2)`

Answer

To solve the expression \( \cos^4 \left( \frac{\pi}{8} \right) + \cos^4 \left( \frac{3\pi}{8} \right) + \cos^4 \left( \frac{5\pi}{8} \right) + \cos^4 \left( \frac{7\pi}{8} \right) \), we can use some trigonometric identities and properties.

### Step 1: Rewrite the terms
Notice that:
\[ \cos\left(\frac{5\pi}{8}\right) = \sin\left(\frac{3\pi}{8}\right) \quad \text{and} \quad \cos\left(\frac{7\pi}{8}\right) = -\sin\left(\frac{\pi}{8}\right) \]
Thus, we can rewrite the expression as:
\[ \cos^4 \left( \frac{\pi}{8} \right) + \sin^4 \left( \frac{3\pi}{8} \right) + \sin^4 \left( \frac{3\pi}{8} \right) + \sin^4 \left( \frac{\pi}{8} \right) \]

### Step 2: Group the terms
We can group the terms:
\[ \left( \cos^4 \left( \frac{\pi}{8} \right) + \sin^4 \left( \frac{\pi}{8} \right) \right) + \left( \cos^4 \left( \frac{3\pi}{8} \right) + \sin^4 \left( \frac{3\pi}{8} \right) \right) \]

### Step 3: Use the identity for \( a^4 + b^4 \)
Recall the identity:
\[ a^4 + b^4 = (a^2 + b^2)^2 - 2a^2b^2 \]
Applying this to our grouped terms:
\[ \cos^4 x + \sin^4 x = (\cos^2 x + \sin^2 x)^2 - 2\cos^2 x \sin^2 x \]
Since \( \cos^2 x + \sin^2 x = 1 \):
\[ \cos^4 x + \sin^4 x = 1 - 2\cos^2 x \sin^2 x \]

### Step 4: Substitute back
Now substituting back into our expression:
\[ \cos^4 \left( \frac{\pi}{8} \right) + \sin^4 \left( \frac{\pi}{8} \right) = 1 - 2\cos^2 \left( \frac{\pi}{8} \right) \sin^2 \left( \frac{\pi}{8} \right) \]
And similarly for \( \frac{3\pi}{8} \):
\[ \cos^4 \left( \frac{3\pi}{8} \right) + \sin^4 \left( \frac{3\pi}{8} \right) = 1 - 2\cos^2 \left( \frac{3\pi}{8} \right) \sin^2 \left( \frac{3\pi}{8} \right) \]

### Step 5: Combine results
Thus, we have:
\[ \cos^4 \left( \frac{\pi}{8} \right) + \sin^4 \left( \frac{\pi}{8} \right) + \cos^4 \left( \frac{3\pi}{8} \right) + \sin^4 \left( \frac{3\pi}{8} \right) = 2 - 2\left( \cos^2 \left( \frac{\pi}{8} \right) \sin^2 \left( \frac{\pi}{8} \right) + \cos^2 \left( \frac{3\pi}{8} \right) \sin^2 \left( \frac{3\pi}{8} \right) \right) \]

### Step 6: Simplify further
Using the double angle identity:
\[ \sin(2x) = 2\sin(x)\cos(x) \]
We can express \( \sin^2 \left( \frac{\pi}{4} \right) = \frac{1}{2} \) and \( \sin^2 \left( \frac{3\pi}{4} \right) = \frac{1}{2} \):
\[ = 2 - 2\left( \frac{1}{4} + \frac{1}{4} \right) = 2 - 2 \cdot \frac{1}{2} = 2 - 1 = 1 \]

### Final Answer
Thus, the final value is:
\[ \boxed{3} \]