What is tan[2 tan⁻¹(1/2) - π/4] equal to?

Let θ = tan⁻¹(1/2), so tan θ = 1/2. tan 2θ = 2(1/2)/(1 - 1/4) = 1/(3/4) = 4/3. Then tan(2θ - π/4) = (tan 2θ - 1)/(1 + tan 2θ) = (4/3 - 1)/(1 + 4/3) = (1/3)/(7/3) = 1/7.