Question

Cos pi/7. cos 2pi/7. cos 4pi/7 = ?

Answer 1

Please see the attachment

Answer 2

Answer:

cos $\frac{\pi }{7}$. cos$\frac{2\pi }{7}$. cos$\frac{4\pi }{7}$ = $\frac{1}{8}$

Step-by-step explanation:

•  The given trigonometric expression,

              cos $\frac{\pi }{7}$. cos$\frac{2\pi }{7}$. cos$\frac{4\pi }{7}$

                = 2sin$\frac{\pi }{7}$.cos$\frac{2\pi }{7}$. cos$\frac{4\pi }{7}$ ×  $\frac{1}{2sin\frac{\pi }{7} }$

                = (2sin$\frac{\pi }{7}$cos$\frac{2\pi }{7}$). cos$\frac{2\pi }{7}$. cos$\frac{4\pi }{7}$ ×  $\frac{1}{2sin\frac{\pi }{7} }$

                = (sin$\frac{2\pi }{7}$). cos$\frac{2\pi }{7}$.cos$\frac{4\pi }{7}$ ×  $\frac{1}{2sin\frac{\pi }{7} }$   [ As we know sin2A = 2sinAcosA]

                = (2sin$\frac{2\pi }{7}$cos$\frac{2\pi }{7}$).cos$\frac{4\pi }{7}$ ×  $\frac{1}{2sin\frac{\pi }{7} }$ × $\frac{1}{2}$

                = sin$\frac{4\pi }{7}$ cos$\frac{4\pi }{7}$ ×  $\frac{1}{2sin\frac{\pi }{7} }$ × $\frac{1}{2}$

                = (2sin$\frac{4\pi }{7}$ cos$\frac{4\pi }{7}$)×  $\frac{1}{2sin\frac{\pi }{7} }$ × $\frac{1}{2}$ × $\frac{1}{2}$

                = sin$\frac{8\pi }{7}$×  $\frac{1}{2sin\frac{\pi }{7} }$ × $\frac{1}{2}$ × $\frac{1}{2}$

                = $\frac{sin\frac{8\pi }{7} }{8sin\frac{\pi }{7} }$.............................(1)

• sin$\frac{8\pi }{7}$ = sin ($\pi +\frac{\pi }{7}$)

                  = sin $\frac{\pi }{7}$         [ As we know sin(π + A) = sinA]

• Putting the value of  sin$\frac{8\pi }{7}$  in equation (1) we get,

             cos $\frac{\pi }{7}$. cos$\frac{2\pi }{7}$. cos$\frac{4\pi }{7}$  = $\frac{sin\frac{8\pi }{7} }{8sin\frac{\pi }{7} }$ = $\frac{sin\frac{\pi }{7} }{8sin\frac{\pi }{7} }$ = $\frac{1}{8}$