Proving that $\sec\frac{\pi}{11}\sec\frac{2\pi}{11}\sec\frac{3\pi}{11}\sec\frac{4\pi}{11}\sec\frac{5\pi}{11}=32$ 
If $11 \gamma = \pi$ then prove that
  $
 \sec(\gamma) \sec(2\gamma) \sec(3\gamma) \sec(4\gamma) \sec(5\gamma) = 32.
$

I could not use the relation $11 \gamma = \pi$.

Let
$$P=\cos\gamma\cos 2\gamma\cos 3\gamma \cos 4\gamma \cos 5\gamma$$
$$Q=\sin\gamma\sin 2\gamma\sin 3\gamma \sin 4\gamma \sin 5\gamma$$
Then, prove that $$2^5PQ=Q$$ using $$2\sin\alpha\cos\alpha=\sin(2\alpha),\quad \sin\beta=\sin(\pi-(11\gamma-\beta))$$

$$\begin{align}2^5PQ&=(2\sin\gamma\cos\gamma)(2\sin 2\gamma\cos 2\gamma)(2\sin 3\gamma\cos 3\gamma)(2\sin 4\gamma \cos 4\gamma)(2\sin 5\gamma \cos 5\gamma)\\&=\sin 2\gamma \sin 4\gamma \sin 6\gamma \sin 8\gamma \sin 10\gamma\\&=\sin 2\gamma \sin 4\gamma \sin (\pi-5\gamma) \sin(\pi-3\gamma)\sin(\pi-\gamma)\\&=\sin 2\gamma \sin 4\gamma \sin 5\gamma \sin 3\gamma \sin \gamma\\&=Q\end{align}$$

