As θ takes every real value, the greatest value of cos(θ − 20°) + cos(θ + 40°) is:

Sum to product: cosP + cosQ = 2·cos((P + Q)/2)·cos((P − Q)/2). Here (P + Q)/2 = θ + 10° and (P − Q)/2 = −30°, so the sum is 2·cos(θ + 10°)·cos30° = √3·cos(θ + 10°). Its greatest value is √3 ≈ 1.732, reached at θ = −10° (equivalently 350°), where the two terms are cos(−30°) + cos30° = 0.866 + 0.866. The tempting answer 2 adds the two individual maxima, but the angles differ by a fixed 60°, so the two cosines can never both be 1 at once; at θ = 0°, for instance, the sum is only 1.706. The amplitude is always 2cos(half the gap), so a gap of 90° would give √2 and a gap of 120° would give 1 — neither matches the 60° gap here.