Question

Prove the following:

`sin  pi^"c"/8 = 1/2sqrt(2 - sqrt(2))`

Answer 1

We know that cos `pi/4 = 1/sqrt(2)`

Also `pi/8`  lies in the first quadrant, hence sin `pi/8` is positive.

Now, cos 2θ = 1 − 2sin² θ

By putting θ = `pi/8`, we get,

`cos  pi/4 = 1 - 2sin^2  pi/8`

∴ `2sin^2  pi/8 = 1 - cos  pi/4`

= `1 - 1/sqrt(2)`

= `(sqrt2-1)/sqrt2`

∴ `sin^2  pi/8 = (sqrt(2) - 1)/(2sqrt(2))`

= `(sqrt(2)(sqrt(2) - 1))/4`

= `(2 - sqrt(2))/4`

∴ `sin  pi/8 = 1/2 sqrt(2-sqrt2)` ......`[∵ sin  pi/8  "is positive"]`