The product tan(90° + θ)·cot(180° − θ) simplifies to:

An angle measured from 90° swaps the ratio for its co-ratio, so tan(90° + θ) = −cotθ; an angle measured from 180° keeps the ratio, so cot(180° − θ) = −cotθ. Both factors are negative, so the product is +co$t^2$θ. Check at θ = 30°: tan120° = −1.732 and cot150° = −1.732, and their product is 3 = co$t^2$30°. Catching only one of the two sign flips gives −co$t^2$θ, treating 180° like 90° gives ta$n^2$θ, and assuming the factors are reciprocals gives 1.