Value of `cos^3 (pi/8)cos^3 ((3pi)/8) + sin^3 (pi/8)sin^3 ((3pi)/8)` is

Options

A. `1/sqrt2`
B. `1/4`
C. `1/(8sqrt2)`
D. `1/2`

Answer

To solve the problem, we need to find the value of the expression:

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

### Step 1: Rewrite \( \frac{3\pi}{8} \)
We can express \( \frac{3\pi}{8} \) as \( \frac{\pi}{2} - \frac{\pi}{8} \).

### Step 2: Use Trigonometric Identities
Using the identity \( \cos\left(\frac{\pi}{2} - \theta\right) = \sin(\theta) \):
\[ \cos\left(\frac{3\pi}{8}\right) = \cos\left(\frac{\pi}{2} - \frac{\pi}{8}\right) = \sin\left(\frac{\pi}{8}\right) \]

### Step 3: Rewrite \( \sin\left(\frac{3\pi}{8}\right) \)
Using the identity \( \sin\left(\frac{\pi}{2} - \theta\right) = \cos(\theta) \):
\[ \sin\left(\frac{3\pi}{8}\right) = \sin\left(\frac{\pi}{2} - \frac{\pi}{8}\right) = \cos\left(\frac{\pi}{8}\right) \]

### Step 4: Substitute Back into the Expression
Now we can substitute these values back into the original expression:
\[ \cos^3\left(\frac{\pi}{8}\right) \sin^3\left(\frac{\pi}{8}\right) + \sin^3\left(\frac{\pi}{8}\right) \cos^3\left(\frac{\pi}{8}\right) \]

This simplifies to:
\[ \cos^3\left(\frac{\pi}{8}\right) \sin^3\left(\frac{\pi}{8}\right) + \sin^3\left(\frac{\pi}{8}\right) \cos^3\left(\frac{\pi}{8}\right) = 2 \cos^3\left(\frac{\pi}{8}\right) \sin^3\left(\frac{\pi}{8}\right) \]

### Step 5: Factor Out the Expression
We can factor this expression:
\[ 2 \cos^3\left(\frac{\pi}{8}\right) \sin^3\left(\frac{\pi}{8}\right) = 2 \left(\cos\left(\frac{\pi}{8}\right) \sin\left(\frac{\pi}{8}\right)\right)^3 \]

### Step 6: Use the Double Angle Identity
Using the identity \( 2 \sin(\theta) \cos(\theta) = \sin(2\theta) \):
\[ \cos\left(\frac{\pi}{8}\right) \sin\left(\frac{\pi}{8}\right) = \frac{1}{2} \sin\left(\frac{\pi}{4}\right) \]

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

### Step 7: Substitute Back
Now substituting back:
\[ = 2 \left(\frac{1}{2} \sin\left(\frac{\pi}{4}\right)\right)^3 = 2 \left(\frac{1}{2} \cdot \frac{1}{\sqrt{2}}\right)^3 \]

### Step 8: Calculate the Final Value
Calculating this gives:
\[ = 2 \cdot \frac{1}{8} \cdot \frac{1}{2\sqrt{2}} = \frac{1}{8\sqrt{2}} \]

### Final Result
Thus, the final value of the expression is:
\[ \frac{1}{8\sqrt{2}} \]