Express tan22.5 using cos45

Cos45 = cos (2*22.5) = (cos22.5) ^2 - (sin22.5) ^2 = (cos22.5) ^2 * (1 - (tan22.5) ^2)

= (1 + cos45) / 2 * (1 - (tan22.5) ^2)

Now, solve for tan22.5:

tan22.5 = sqrt (1 - 2cos45 / (1+cos45))