${\mathrm{cosec}}^2 {\mathrm{Acot}}^2 \mathrm{A}-{\sec }^2 {\mathrm{Atan}}^2 \mathrm{A}-\left({\cot }^2 \mathrm{A}-{\tan }^2 \mathrm{A}\right)$ $\left({\sec }^2 {\mathrm{Acosec}}^2 \mathrm{A}-1\right)=$

$\begin{array}{rcl} & & {\mathrm{cosec}}^2\mathrm{A} {\cot }^2\mathrm{A}-{\sec }^2\mathrm{A} {\tan }^2\mathrm{A}-\left({\cot }^2\mathrm{A}-{\tan }^2\mathrm{A}\right) \left({\sec }^2\mathrm{A} {\mathrm{cosec}}^2\mathrm{A}-1\right)\\ & & -\left(1+{\cot }^2\mathrm{A}\right) {\cot }^2\mathrm{A}-\left(1-{\tan }^2\mathrm{A}\right) {\tan }^2\mathrm{A}-\\ & & \left({\cot }^2\mathrm{A}-{\tan }^2\mathrm{A}\right) \left(\left(1+{\tan }^2\mathrm{A}\right) \left(1+{\cot }^2\mathrm{A}\right)-1\right)\\ & =& {\cot }^2\mathrm{A}+{\cot }^4\mathrm{A}-{\tan }^2\mathrm{A}-{\tan }^4\mathrm{A}\\ & & -\left({\cot }^2\mathrm{A}-{\tan }^2\mathrm{A}\right) \left(1+{\tan }^2\mathrm{A}+{\cot }^2\mathrm{A}\right)\\ & =& {\cot }^2\mathrm{A}-{\tan }^2\mathrm{A}+{\cot }^4\mathrm{A}-{\tan }^4\mathrm{A}-\left({\cot }^2\mathrm{A}-{\tan }^2\mathrm{A}\right)\\ & & -1+{\tan }^4\mathrm{A}-{\cot }^4\mathrm{A}+1\\ & =& 0\end{array}$