If cosecθ + cotθ = 11 and θ is acute, then cosθ equals:

Since cose$c^2$θ − co$t^2$θ = 1, the two expressions are reciprocals, so cosecθ − cotθ = 1/11. Adding gives 2cosecθ = 11 + 1/11 = 122/11, so cosecθ = 61/11 and sinθ = 11/61; subtracting gives 2cotθ = 11 − 1/11 = 120/11, so cotθ = 60/11. The right triangle is therefore 11, 60, 61 (and $11^2+60^2=3721=61^2$), giving cosθ = 60/61 ≈ 0.9836, with θ ≈ 10.4° — a small angle, as an 11 in the sum demands. The near misses are the other ratios of that same triangle: 11/61 is sinθ and 61/60 is secθ, while 1/11 is just the conjugate expression cosecθ − cotθ.