Let's salute cos 1

$=\cos \frac{\pi}{11} \cos \frac{2\pi}{11} \cos \frac{3\pi}{11} \cos \frac{4\pi}{11} \cos \frac{5\pi}{11} \\ =\frac{2 \sin \frac{\pi}{11}}{2 \sin \frac{\pi}{11}} \cos \frac{\pi}{11} \cos \frac{2\pi}{11} \cos \frac{3\pi}{11} \cos \frac{4\pi}{11} \cos \frac{5\pi}{11} \\ \text{Now, } 2 \sin \frac{\pi}{11} \cos \frac{\pi}{11}=\sin \frac{2\pi}{11} \text{ because } \color{#D61F06}{2 \sin A \cos A=\sin 2A} \\ \text{Note: We are going to use this identity again and again.} \\ =\frac{2}{4 \sin \frac{\pi}{11}} \sin \frac{2\pi}{11} \cos \frac{2\pi}{11} \cos \frac{3\pi}{11} \cos \frac{4\pi}{11} \cos \frac{5\pi}{11} \\ =\frac{2}{8 \sin \frac{\pi}{11}} \sin \frac{4\pi}{11} \cos \frac{4\pi}{11} \cos \frac{3\pi}{11} \cos \frac{5\pi}{11} \\ =\frac{1}{8 \sin \frac{\pi}{11}} \color{#3D99F6}{ \sin \frac{8\pi}{11}} \cos \frac{3\pi}{11} \cos \frac{5\pi}{11} \\ \text{Note: } \color{#3D99F6}{\sin \frac{8\pi}{11}}=\sin \left(\pi-\frac{3\pi}{11}\right)=\sin \frac{3\pi}{11} \\ =\frac{2}{16 \sin \frac{\pi}{11}} \sin \frac{3\pi}{11} \cos \frac{3\pi}{11} \cos \frac{5\pi}{11} \\ =\frac{1}{16 \sin \frac{\pi}{11}} \color{#EC7300}{\sin \frac{6\pi}{11}} \cos \frac{5\pi}{11} \\ \text{Note: } \color{#EC7300}{\sin \frac{6\pi}{11}}=\sin \left(\pi-\frac{5\pi}{11}\right)=\sin \frac{5\pi}{11} \\ =\frac{2}{32 \sin \frac{\pi}{11}} \sin \frac{5\pi}{11} \cos \frac{5\pi}{11} \\ =\frac{1}{32 \sin \frac{\pi}{11}}\color{#20A900}{ \sin \frac{10\pi}{11}} \\ \text{Note: } \color{#20A900}{sin \frac{10\pi}{11}}=sin \left(\pi-\frac{\pi}{11}\right)=\sin \frac{\pi}{11} \\ =\frac{1}{32 \sin \frac{\pi}{11}} \sin \frac{\pi}{11}=\frac{1}{32} \\ \text{Therefore, }A=\boxed{32}$

Let $\displaystyle P=\cos \frac{\pi}{11}\cos \frac{2\pi}{11}\cos \frac{3\pi}{11}\cos \frac{4\pi}{11}\cos \frac{5\pi}{11}$ and $\displaystyle Q=\sin \frac{\pi}{11}\sin \frac{2\pi}{11}\sin \frac{3\pi}{11}\sin \frac{4\pi}{11}\sin \frac{5\pi}{11}$ .

Then $\displaystyle 32PQ =\sin \frac{2\pi}{11}\sin \frac{4\pi}{11}\sin \frac{6\pi}{11}\sin \frac{8\pi}{11}\sin \frac{10\pi}{11}=Q$ and hence $\displaystyle P =\frac{1}{32}$ .