If cosec(3θ + 10°) = sec(θ + 20°) and both of these angles are acute, then θ equals:

A cosecant equals a secant when the two angles are complementary, because cosec(90° − $x$) = secx. So (3θ + 10°) + (θ + 20°) = 90°, giving 4θ + 30° = 90° and θ = 15°. Check: the angles become 55° and 35°, and cosec55° = sec35° = 1.2208. Setting the two angles equal to each other instead — the commonest wrong move, since cosec and sec look like a matching pair — gives θ = 5°, where cosec25° = 2.366 but sec25° = 1.103.