Cosine problem which relate to sine

$\begin{aligned} P & = 63 \sec \frac \pi{33} \sec \frac {2\pi}{33} \sec \frac {4\pi}{33} \sec \frac {8\pi}{33} \sec \frac {16\pi}{33} \\ & = \frac {63}{\cos \frac \pi{33} \cos \frac {2\pi}{33} \cos \frac {4\pi}{33} \cos \frac {8\pi}{33} \cos \frac {16\pi}{33}} \\ & = \frac {63 \sin \frac \pi{33} }{\sin \frac \pi{33}\cos \frac \pi{33} \cos \frac {2\pi}{33} \cos \frac {4\pi}{33} \cos \frac {8\pi}{33} \cos \frac {16\pi}{33}} \\ & = \frac {63 \cdot 2 \sin \frac \pi{33} }{\sin \frac {2\pi}{33} \cos \frac {2\pi}{33} \cos \frac {4\pi}{33} \cos \frac {8\pi}{33} \cos \frac {16\pi}{33}} \\ & = \frac {63 \cdot 2^2 \sin \frac \pi{33}}{\sin \frac {4\pi}{33} \cos \frac {4\pi}{33} \cos \frac {8\pi}{33} \cos \frac {16\pi}{33}} \\ & = \frac {63 \cdot 2^3 \sin \frac \pi{33}}{\sin \frac {8\pi}{33} \cos \frac {8\pi}{33} \cos \frac {16\pi}{33}} \\ & = \frac {63 \cdot 2^4 \sin \frac \pi{33}}{\sin \frac {16\pi}{33} \cos \frac {16\pi}{33}} \\ & = \frac {63 \cdot 2^5 \sin \frac \pi{33}}{\color{#3D99F6}\sin \frac {32\pi}{33}} & \small {\color{#3D99F6}\text{Note that }\sin (\pi-x) = \sin x} \\ & = \frac {63 \cdot 2^5 \sin \frac \pi{33}}{\color{#3D99F6}\sin \frac {\pi}{33}} \\ & = 63 \cdot 2^5 = 63 \cdot 32 = \boxed{2016} \end{aligned}$