If cosecθ − sinθ = $a^3$ and secθ − cosθ = $b^3$, then $a^2{b}^2(a^2+b^2)$ equals:

cosecθ − sinθ = $co$$s^2$θ/sinθ and secθ − cosθ = $si$$n^2$θ/cosθ, so $a=(co{s}^2\theta /sin\theta {)}^{1/3}$ and $b=(si{n}^2\theta /cos\theta {)}^{1/3}$. Then $a^2$$b^2=(sin\theta \cdot cos\theta {)}^{2/3}$, while $a^2+b^2=(co$$s^2$θ + $si$$n^2$θ)/$(sin\theta \cdot cos\theta {)}^{2/3}=1/(sin\theta \cdot cos\theta {)}^{2/3}$. Multiplying the two results gives 1.