Question

Use a half-angle identity to find the exact value of tan pi/8

Answer 1

what you need to show, just using the formula you know, or how to get the formula itself, how much

Answer 2

\[\tan 2x=\frac{ 2\tan x }{ 1-\tan ^2x }\]
if \[x=\frac{ \pi }{ 8 },2x=2 \times \frac{ \pi }{ 8 }=\frac{ \pi }{ 4 },\tan 2x=\tan \frac{ \pi }{ 4 }=1\]
\[1=\frac{ 2\tan x }{ 1-\tan ^2x },1-\tan ^2x=2\tan x\]
\[\tan ^2x+2\tan x-1=0\]
\[\tan x=\frac{ -1\pm \sqrt{(2)^2-4*1*-1} }{ 2*1 }=\frac{ -1\pm \sqrt{8} }{ 2 }\]
\[\tan \frac{ \pi }{ 8 }~is~positive.\]
taking positive sign only.
\[x=\frac{ \pi }{ 4 }\]
\[\tan \frac{ \pi }{ 4 }=\frac{ -1+\sqrt{8} }{ 2 }\]

Answer 3

Thanx, I already got the answer. It was \[\sqrt{2}-1\]