For every θ with 0° < θ < 90°, the value of sinθ + cosθ satisfies:

$(sin\theta +cos\theta {)}^2=1+2$sinθ·cosθ = 1 + sin2θ. For 0° < θ < 90° the value of sin2θ lies in (0, 1], so the square lies in (1, 2] and the sum lies in (1, √2]. The peak √2 ≈ 1.414 happens at θ = 45°, and near the ends the sum sinks towards 1 without ever reaching it — so the sum is always more than either ratio alone.