Question

Prove the following:

`tan  pi/8 = sqrt(2) - 1`

Answer 1

We know that,

tan 2θ = `(2tantheta)/(1 - tan^2theta)`

By putting θ = `pi/8`, we get

`tan  pi/4  = (2tan  pi/8)/(1 - tan^2  pi/8)`

Let `tan  pi/8` = x.

Then 1 = `(2x)/(1 - x^2)`

∴ 1 – x² = 2x

∴ x² + 2x – 1 = 0

∴ x = `(-2 ± sqrt(4 - 4(1)(-1)))/(2 xx 1)`

= `(-2 ± sqrt(8))/2`

= `(-2 ± 2sqrt(2))/2`

= `-1 ± sqrt(2)`

Since `pi/8` lies in the first quadrant, x = `tan  pi/8` is positive.

∴ `tan  pi/8 = sqrt(2) - 1`.