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Question
Prove the following:
`tan pi/8 = sqrt(2) - 1`
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Solution
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`.
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