Question

Evaluate: \( (1 + \cos\frac{\pi}{8})(1 + \cos\frac{2\pi}{8})(1 + \cos\frac{3\pi}{8}) \ldots (1 + \cos\frac{7\pi}{8}) = \)

• A. \( \frac{1}{16} \)
• B. \( \frac{1}{64} \)
• C. \( \frac{3}{16} \)
• D. \( \frac{3}{64} \)

Hint:

When evaluating trigonometric products of cosine angles in arithmetic progression, check for known product identities involving symmetry.

Answer 1

The Correct Option is A

Use the known product identity:
\[ \prod_{k=1}^{n-1}(1 + \cos\frac{k\pi}{n}) = \frac{n}{2^{n-1}}, \quad \text{for even } n \] Here, \( n = 8 \), so: \[ \prod_{k=1}^{7}(1 + \cos\frac{k\pi}{8}) = \frac{8}{2^7} = \frac{8}{128} = \boxed{\frac{1}{16}} \]