tan(22.5 degree)

As follows:

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/May07/Crazytrigo54.jpg

tan45=1

tan(2X22.5)=1

2tan22.5/[1-(tan22.5)^2]=1

2tan22.5=1-(tan22.5)^2

(tan22.5)^2+tan22.5-1=0

設tan22.5=X
X^2+2x-1=0

x=√2-1 或 x=-1-√2(捨去)

therefore,tan22.5=√2-1

http://img411.imageshack.us/img411/9100/anglenp8.jpg

Assume the case as shown in the figure above

tan45°=AC/CD=1
AC^2+CD^2=AD^2(Pyth. thereom)
1^2+1^2=√2
∵∠BAD=∠ADC-∠ABD(ext.∠ of △)
=45°-22.5°=22.5°
∠ABD=∠BAD=22.5°
∴△ABD is isos.△(sides opp. eq.∠s)
BD=AD=√2
tan22.5°=1/ (1+√2)

(Square root 2)-1

You can draw a graph with a square and a rcetangle.
The lenght of the square can let be 1.
The lenghts of the rectangle can let be 1 and Square root 2.