Question

The value of `2sin(π/8)sin((2π)/8)sin((3π)/8)sin((5π)/8)sin((6π)/8)sin((7π)/8)` is ______.

Options

• `1/(4sqrt(2)`

• `1/4`

• `1/8`

• `1/(8sqrt(2)`

Answer 1

The value of `2sin(π/8)sin((2π)/8)sin((3π)/8)sin((5π)/8)sin((6π)/8)sin((7π)/8)` is `underlinebb(1/8)`.

Explanation:

We know

`sin  (5π)/8 = sin(π - (3π)/8) = sin((3π)/8)`  ...(i)

`sin  (7π)/8 = sin(π - π/8) = sin  π/8`  ...(ii)

`sin  (2π)/8 = sin  π/4 = 1/sqrt(2)`  ...(iii)

`sin  (6π)/8 = sin ((3π)/8) = 1/sqrt(2)`  ...(iv)

`2sin(π/8)sin((2π)/8)sin((3π)/8)sin((5π)/8)sin((6π)/8)sin((7π)/8)`

= `2sin^2  π/8 xx sin^2((3π)/8) xx (1/sqrt(2)) xx (1/sqrt(2))`

= `2sin^2  π/8 xx sin^2((3π)/8) xx 1/2`

= `(sin  π/8. cos  π/8)^2`  ...`[∵ sin  (3π)/8 = sin(π/2 - π/8) = cos  π/8]`

= `(1/2 sin  π/4)^2`

= `(1/(2sqrt(2)))^2`

= `1/8`