Question

The value of the following expression is: \[ \cos \left( \frac{\pi}{2^2} \right) \cdot \cos \left( \frac{\pi}{2^3} \right) \cdot \cos \left( \frac{\pi}{2^4} \right) ... \cdot \cos \left( \frac{\pi}{2^{10}} \right) \]

• A. \( \frac{\sin\left( \frac{\pi}{2^{10}} \right)}{512} \)
• B. \( \frac{\csc\left( \frac{\pi}{2^{10}} \right)}{512} \)
• C. \( \frac{\sin\left( \frac{\pi}{2^{10}} \right)}{1024} \)
• D. \( \frac{\csc\left( \frac{\pi}{2^{10}} \right)}{1024} \)

Hint:

For products of cosines, the formula \( \prod_{k=1}^{n} \cos\left( \frac{\pi}{2^k} \right) = \frac{\sin\left( \frac{\pi}{2^n} \right)}{2^{n-1}} \) is useful.

Answer 1

The Correct Option is B

The product of cosines follows a known formula: \[ \prod_{k=1}^{n} \cos\left( \frac{\pi}{2^k} \right) = \frac{\sin\left( \frac{\pi}{2^n} \right)}{2^{n-1}} \] For \( n = 10 \), this becomes: \[ \prod_{k=1}^{10} \cos\left( \frac{\pi}{2^k} \right) = \frac{\sin\left( \frac{\pi}{2^{10}} \right)}{2^{9}} = \frac{\sin\left( \frac{\pi}{2^{10}} \right)}{512} \] Thus, the answer is \( \frac{\csc\left( \frac{\pi}{2^{10}} \right)}{512} \).