Question

Let \[ \alpha= \left(1-2\cos\frac{\pi}{11}\right) \left(1-2\cos\frac{3\pi}{11}\right) \left(1-2\cos\frac{9\pi}{11}\right) \left(1-2\cos\frac{27\pi}{11}\right) \left(1-2\cos\frac{81\pi}{11}\right) \] Then the value of \[ 5-\alpha^2 \] is ________.

Hint:

Products involving: \[ 1-2\cos\theta \] are often simplified using: \[ z=e^{i\theta} \] and roots of unity identities.

Answer 1

Correct Answer: 4