Question

I'm confused! Please explain!

Answer 1

First, use the fact that
`\tan x` is
`\pi`-periodic. This means
`\tan(x+\pi)=\tan x` for all
`x`, and in particular

`\tan\frac{9\pi}8=\tan\left(\frac\pi8+\pi\right)=\tan\frac\pi8`

Now, recall the half-angle identities,

`\sin^2\frac x2=\frac{1-\cos x}2`

`\cos^2\frac x2=\frac{1+\cos x}2`

`\implies\tan^2\frac x2=(\sin^2\frac x2)/(\cos^2\frac x2)=(1-\cos x)/(1+\cos x)`

When we take the square root, we get two possible values, but we know
`\tan x>0` for
`0<x<\frac\pi2`, so we know to take the positive square root:

`\tan\frac x2=\sqrt{(1-\cos x)/(1+\cos x)}`

So we have

`\tan\frac\pi8=\sqrt{(1-\cos\frac\pi4)/(1+\cos\frac\pi4)}=\sqrt{\frac{1-\frac1{\sqrt2}}{1+\frac1{\sqrt2}}}=√((\sqrt2-1)^2)`

`\boxed{\tan\frac\pi8=\sqrt2-1}`