If f(θ) = 1/(1 + tanθ) and α + β = 5π/4, then what is f(α)f(β)?

f(α)f(β) = 1/[(1+tanα)(1+tanβ)] = 1/[1 + tanα + tanβ + tanα tanβ]. Since α + β = 5π/4, tan(α+β) = tan(5π/4) = 1. So (tanα + tanβ)/(1 – tanα tanβ) = 1, giving tanα + tanβ = 1 – tanα tanβ. Substituting: denominator = 1 + (1 – tanα tanβ) + tanα tanβ = 2. So f(α)f(β) = 1/2.