Solve? You mean looking it up on a calculator is not enough?

Anyway, my calculator here says tan(pi/8) = 0.414213562.

Or, maybe, you mean this, (one of many ways):

tan(pi/8) = tan[(pi/4)/2]

We know the trig functions value of pi/4 (or 45 degrees) without bothering to consult a clculator, so, maybe this is what you mean by "solve for tan(pi/8).

Okay. Then we know also this trig identity:

tan(A/2) = (1 -cosA)/sinA = sinA/(1 +cosA) ----***

If A = pi/4, then without calculator, we can solve for tan(A/2).

tan(pi/8)

= tan[(pi/4)/2]

= [1 -cos(pi/4)]/[sin(pi/4)]

= [ 1 -1/sqrt(2)]/[1/sqrt(2)]

= in the end, sqrt(2) -1

= 1.4142... -1

= 0.4142.....

Or,

tan(pi/8)

= [sin(pi/4)]/[1 +cos(pi/4)]

= sqrt(2) -1

= 0.4142....