Using the half-angle formula tan(θ/2) = (1 − cosθ)/sinθ with θ = 45°, the exact value of tan22.5° is:

With cos45° = sin45° = √2/2, the formula gives (1 − √2/2)/(√2/2) = (2 − √2)/√2 = √2 − 1 ≈ 0.414. It must be positive and less than 1 because 22.5° is below 45°, which rules out 1 − √2 ≈ −0.414 and √2 + 1 ≈ 2.414 (that one is cot22.5°).