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

f(α)·f(β) = 1/[(1 + tanα)(1 + tanβ)]. Expand: (1 + tanα)(1 + tanβ) = 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: 1 + (1 – tanα·tanβ) + tanα·tanβ = 2. Therefore f(α)·f(β) = 1/2.