If α + β = 45° and (tanα+1)(tanβ+1) = 2x, then x is:

Since α+β = 45°, tan(α+β) = 1, so (tanα+tanβ)/(1−tanα·tanβ) = 1, meaning tanα+tanβ = 1−tanα·tanβ. Expanding (tanα+1)(tanβ+1) = tanα·tanβ + tanα + tanβ + 1 = tanα·tanβ + (1−tanα·tanβ) + 1 = 2. So 2x = 2, giving x = 1.