Question

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

• A. \( \frac{1}{2\sqrt{2}} \)
• B. \( \frac{1}{2} \)
• C. \( \frac{1}{\sqrt{2}} \)
• D. \( \frac{1}{4} \)

Hint:

Use power reduction formulas to simplify expressions involving higher powers of sine and cosine functions.

Answer 1

The Correct Option is A

The given expression can be rewritten using trigonometric identities:
\[ \cos^3 A \cos A + \sin^3 A \sin A = \cos A \sin A \left( \cos^2 A - \sin^2 A \right) + \cos^4 A + \sin^4 A. \]
Using the identity:
\[ \cos^4 A + \sin^4 A = \frac{1}{2} + \frac{1}{2} \cos 4A \]
and substituting \( A = \frac{3\pi}{8} \), we calculate step-by-step:
1. Compute \( \cos 4A = \cos \frac{12\pi}{8} = \cos \frac{3\pi}{2} = 0 \).
2. Using \( \cos^4 A + \sin^4 A = \frac{1}{2} \).
3. Calculate \( \cos A \sin A = \frac{1}{2} \sin 2A \).
4. Compute \( \sin 2A = \sin \frac{6\pi}{8} = \sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2} \).
5. Multiply: \( \frac{1}{2} \times \frac{\sqrt{2}}{2} = \frac{1}{2\sqrt{2}} \).
Thus, the final answer is:
\[ \boxed{\frac{1}{2\sqrt{2}}}. \]