Question

prove that Cosπ/17.Cos2π/17.cos4π/17.cos8π/17=1/16​

Answer 1

Step-by-step explanation:

We have,

$\sf{cos \left( \dfrac{\pi}{17} \right) \cdot \: cos \left( \dfrac{2\pi}{17} \right) \cdot \:cos \left( \dfrac{4\pi}{17} \right) \cdot \: cos \left( \dfrac{8\pi}{17} \right) }$

$\sf{ = \dfrac{1}{2 \: sin\left( \dfrac{\pi}{17} \right) } \cdot \: \left \{2 \: sin\left( \dfrac{\pi}{17} \right) cos \left( \dfrac{\pi}{17} \right) \right\}\cdot \: cos \left( \dfrac{2\pi}{17} \right) \cdot \:cos \left( \dfrac{4\pi}{17} \right) \cdot \: cos \left( \dfrac{8\pi}{17} \right) }$

$\sf{ = \dfrac{1}{2 \: sin\left( \dfrac{\pi}{17} \right) } \cdot \: sin\left( \dfrac{2\pi}{17} \right) \cdot \: cos \left( \dfrac{2\pi}{17} \right) \cdot \:cos \left( \dfrac{4\pi}{17} \right) \cdot \: cos \left( \dfrac{8\pi}{17} \right) }$

$\sf{ = \dfrac{1}{4 \: sin\left( \dfrac{\pi}{17} \right) } \cdot 2\: sin\left( \dfrac{2\pi}{17} \right) \: cos \left( \dfrac{2\pi}{17} \right) \cdot \:cos \left( \dfrac{4\pi}{17} \right) \cdot \: cos \left( \dfrac{8\pi}{17} \right) }$

$\sf{ = \dfrac{1}{4 \: sin\left( \dfrac{\pi}{17} \right) } \cdot \: sin\left( \dfrac{4\pi}{17} \right) \cdot \:cos \left( \dfrac{4\pi}{17} \right) \cdot \: cos \left( \dfrac{8\pi}{17} \right) }$

$\sf{ = \dfrac{1}{8 \: sin\left( \dfrac{\pi}{17} \right) } \cdot 2\: sin\left( \dfrac{4\pi}{17} \right) cos \left( \dfrac{4\pi}{17} \right) \cdot \: cos \left( \dfrac{8\pi}{17} \right) }$

$\sf{ = \dfrac{1}{8 \: sin\left( \dfrac{\pi}{17} \right) } \cdot \: sin\left( \dfrac{8\pi}{17} \right) \cdot \: cos \left( \dfrac{8\pi}{17} \right) }$

$\sf{ = \dfrac{1}{16 \: sin\left( \dfrac{\pi}{17} \right) } \cdot 2\: sin\left( \dfrac{8\pi}{17} \right) cos \left( \dfrac{8\pi}{17} \right) }$

$\sf{ = \dfrac{1}{16 \: sin\left( \dfrac{\pi}{17} \right) } \cdot sin\left( \dfrac{16\pi}{17} \right) }$

$\sf{ = \dfrac{1}{16 \: sin\left( \dfrac{\pi}{17} \right) } \cdot sin\left( \pi - \dfrac{\pi}{17} \right) }$

$\sf{ = \dfrac{1}{16 \: sin\left( \dfrac{\pi}{17} \right) } \cdot sin\left( \dfrac{\pi}{17} \right) }$

$\sf{ = \dfrac{1}{16 } }$