sin(θ) + cos(θ) = 1
Divide both sides by √2:
1/√2 sin(θ) + 1/√2 cos(θ) = 1/√2
We do this because sin(x) = cos(x) = 1/√2 for x = π/4, and this lets us condense the left side using either of the following angle sum identities:
sin(x + y) = sin(x) cos(y) + cos(x) sin(y)
cos(x - y) = cos(x) cos(y) - sin(x) sin(y)
Depending on which identity you choose, we get either
1/√2 sin(θ) + 1/√2 cos(θ) = sin(θ + π/4)
or
1/√2 sin(θ) + 1/√2 cos(θ) = cos(θ - π/4)
Let's stick with the first equation, so that
sin(θ + π/4) = 1/√2
θ + π/4 = π/4 + 2nπ or θ + π/4 = 3π/4 + 2nπ
(where n is any integer)
θ = 2nπ or θ = π/2 + 2nπ
We get only one solution from the second solution set in the interval 0 < θ < 2π when n = 0, which gives θ = π/2.