For every θ at which every factor is defined, (cosecθ − sinθ)(secθ − cosθ)(tanθ + cotθ) equals:

Rewrite each bracket over a single denominator: cosecθ − sinθ = (1 − si$n^2$θ)/sinθ = co$s^2$θ/sinθ, secθ − cosθ = si$n^2$θ/cosθ, and tanθ + cotθ = (si$n^2$θ + co$s^2$θ)/(sinθ·cosθ) = 1/(sinθ·cosθ). Multiplying, the first two give sinθ·cosθ, and the third is its reciprocal, so the product is 1 for every admissible θ. Check at θ = 50°: $0.53937\times 0.91293\times 2.03085=1.0000$. Stopping after two brackets gives sinθ·cosθ; forgetting to invert the third bracket leaves tanθ + cotθ; and 0 would need one factor to vanish, which none does for 0° < θ < 90°.