The value of `cos^(3) ((pi)/(8)). cos((3pi)/(8)) + sin^(3) ((pi)/(8)).sin ((3pi)/(8))` is

Options

A. `1/2sqrt2`
B. `1/sqrt2`
C. `1/2`
D. `-1/2`

Explanation

To solve the expression \( \cos^3 \left( \frac{\pi}{8} \right) \cos \left( \frac{3\pi}{8} \right) + \sin^3 \left( \frac{\pi}{8} \right) \sin \left( \frac{3\pi}{8} \right) \), we will follow these steps:

### Step 1: Rewrite \( \cos \left( \frac{3\pi}{8} \right) \) and \( \sin \left( \frac{3\pi}{8} \right) \)

Using the identity \( \cos \left( \frac{3\pi}{8} \right) = \sin \left( \frac{\pi}{2} - \frac{3\pi}{8} \right) = \sin \left( \frac{\pi}{8} \right) \) and \( \sin \left( \frac{3\pi}{8} \right) = \cos \left( \frac{\pi}{2} - \frac{3\pi}{8} \right) = \cos \left( \frac{\pi}{8} \right) \), we can rewrite the expression:

\[ \cos^3 \left( \frac{\pi}{8} \right) \sin \left( \frac{\pi}{8} \right) + \sin^3 \left( \frac{\pi}{8} \right) \cos \left( \frac{\pi}{8} \right) \]

### Step 2: Factor the expression

Now we can factor out \( \cos \left( \frac{\pi}{8} \right) \sin \left( \frac{\pi}{8} \right) \):

\[ \cos \left( \frac{\pi}{8} \right) \sin \left( \frac{\pi}{8} \right) \left( \cos^2 \left( \frac{\pi}{8} \right) + \sin^2 \left( \frac{\pi}{8} \right) \right) \]

### Step 3: Simplify using Pythagorean identity

Using the Pythagorean identity \( \cos^2 A + \sin^2 A = 1 \):

\[ \cos \left( \frac{\pi}{8} \right) \sin \left( \frac{\pi}{8} \right) \cdot 1 = \cos \left( \frac{\pi}{8} \right) \sin \left( \frac{\pi}{8} \right) \]

### Step 4: Use the double angle formula

We can use the double angle formula for sine, which states that \( \sin(2A) = 2 \sin(A) \cos(A) \):

\[ \sin \left( \frac{\pi}{4} \right) = 2 \sin \left( \frac{\pi}{8} \right) \cos \left( \frac{\pi}{8} \right) \]

Since \( \sin \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \):

\[ \cos \left( \frac{\pi}{8} \right) \sin \left( \frac{\pi}{8} \right) = \frac{1}{2} \sin \left( \frac{\pi}{4} \right) = \frac{1}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{4} \]

### Final Answer

Thus, the value of the expression is:

\[ \frac{\sqrt{2}}{4} \]