If 20cotθ = 21 and θ is acute, then (secθ − cosecθ)/(secθ + cosecθ) equals:

cotθ = 21/20 puts the adjacent side at 21 and the opposite at 20, so the hypotenuse is √(441 + 400) = 29, giving sinθ = 20/29 and cosθ = 21/29. Multiplying numerator and denominator by sinθ·cosθ turns secθ into sinθ and cosecθ into cosθ, so the expression is (sinθ − cosθ)/(sinθ + cosθ) = (20 − 21)/(20 + 21) = −1/41. Swapping the two reciprocals gives +1/41, and inverting the finished fraction gives ∓41.