Question

# The value of $$\cos{\displaystyle\frac{\pi}{7}}\cos{\displaystyle\frac{2\pi}{7}}\cos{\displaystyle\frac{3\pi}{7}}$$ is

A
0
B
12
C
13
D
18

Solution

## The correct option is D $$\displaystyle\frac{1}{8}$$The value of $$\displaystyle \cos { \frac { \pi }{ 7 } } \cos { \frac { 2\pi }{ 7 } } \cos { \frac { 3\pi }{ 7 } }$$$$\displaystyle =\frac { 2\sin { \frac { \pi }{ 7 } } \cos { \frac { \pi }{ 7 } } \cos { \frac { 2\pi }{ 7 } } \cos { \frac { 3\pi }{ 7 } } }{ 2\sin { \frac { \pi }{ 7 } } }$$$$\displaystyle =\frac { \sin { \frac { 2\pi }{ 7 } } \cos { \frac { 2\pi }{ 7 } } \cos { \frac { 3\pi }{ 7 } } }{ 2\sin { \frac { \pi }{ 7 } } }$$$$\displaystyle =\frac { \sin { \frac { 4\pi }{ 7 } } \cos { \frac { 3\pi }{ 7 } } }{ 4\sin { \frac { \pi }{ 7 } } } =\frac { \sin { \frac { 3\pi }{ 7 } } \cos { \frac { 3\pi }{ 7 } } }{ 4\sin { \frac { \pi }{ 7 } } }$$$$\displaystyle =\frac { \sin { \frac { 6\pi }{ 7 } } }{ 8\sin { \frac { \pi }{ 7 } } } =\frac { \sin { \left( \pi -\frac { \pi }{ 7 } \right) } }{ 8\sin { \frac { \pi }{ 7 } } } =\frac { \sin { \frac { \pi }{ 7 } } }{ 8\sin { \frac { \pi }{ 7 } } } =\frac { 1 }{ 8 }$$Mathematics

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