2(22)(322)(522)(722)(922) is equal to:

Question

$2\sin\left(\dfrac\pi{22}\right)\sin\left(\dfrac{3\pi}{22}\right)\sin\left(\dfrac{5\pi}{22}\right)\sin\left(\dfrac{7\pi}{22}\right)\sin\left(\dfrac{9\pi}{22}\right)$ is equal to:

Accepted Answer

$$2\sin {\pi  \over {22}}\sin {{3\pi } \over {22}}\sin {{5\pi } \over {22}}\sin {{7\pi } \over {22}}\sin {{9\pi } \over {22}}$$

$$ = 2\sin \left( {{{11\pi  - 10\pi } \over {22}}} \right)\sin \left( {{{11\pi  - 8\pi } \over {22}}} \right)\sin \left( {{{11\pi  - 6\pi } \over {22}}} \right)\sin \left( {{{11\pi  - 4\pi } \over {22}}} \right)\sin \left( {{{11\pi  - 2\pi } \over {22}}} \right)$$

$$ = 2\cos {\pi  \over {11}}\cos {{2\pi } \over {11}}\cos {{3\pi } \over {11}}\cos {{4\pi } \over {11}}\cos {{5\pi } \over {11}}$$

$$ = {{2\sin {{32\pi } \over {11}}} \over {{2^5}\sin {\pi  \over {11}}}}$$

$$ = {1 \over {16}}$$

Solutions · 1