$\cos e{c}^{2}A{\cot }^{2}A-{\sec }^{2}A{\tan }^{2}A-\left({\cot }^{2}A-{\tan }^{2}A\right)\left({\sec }^{2}A\cos e{c}^{2}A-1\right)=$

$\cos e{c}^{2}45°={\sec }^{2}45°{\tan }^{2}45°-\left({\cot }^{2}45-{\tan }^{2}45°\right)$

$\left({\sec }^{2}45°\cos c{e}^{2}45°-1\right)$

$=2\left(1\right)-2\left(1\right)-\left(1-1\right)$  $\left(2\left(2\right)-1\right)$

$=0$

$\frac{1}{{\sin }^{2}A}\cdot \frac{{\cos }^{2}A}{{\sin }^{2}A}-\frac{1}{{\cos }^{2}A}\cdot \frac{{\sin }^{2}A}{{\cos }^{2}A}-\left(\frac{\cos A}{{\sin }^{2}A}-\frac{\sin A}{{\cos }^{2}A}\right)\left(\frac{1}{{\cos }^{2}A}\cdot \frac{1}{{\sin }^{2}A}-1\right)$

$\frac{{\cos }^{2}A}{{\sin }^{4}A}-\frac{{\sin }^{2}A}{{\cos }^{4}A}-\left(\frac{{\cos }^4-{\sin }^{4}A}{{\sin }^{2}A\cdot {\cos }^{2}A}\right)\left(\frac{1-{\cos }^{2}A\cdot {\sin }^{2}A}{{\sin }^{2}A\cdot {\cos }^{2}A}\right)$

$\frac{{\cos }^{6}A-{\sin }^{6}A}{{\sin }^{4}A\cdot {\cos }^{4}A}-\frac{{\cos }^{4}A-{\sin }^{4}A-{\cos }^{6}A\cdot {\sin }^{2}A+{\cos }^{2}A\cdot {\sin }^{4}A}{{\sin }^{2}A\cdot {\cos }^{2}A}$

$=\frac{{\left({\cos }^{2}A\right)}^3-{\left({\sin }^{2}A\right)}^3}{{\sin }^{4}A\cdot {\cos }^{4}A}-\left[\frac{{\left({\cos }^{2}A\right)}^2-{\left({\sin }^{2}A\right)}^2-{\cos }^{2}A\cdot {\sin }^{2}A\left({\cos }^{4}A-{\sin }^{4}A\right)}{{\sin }^{4}A\cdot {\cos }^{4}A}\right]$

$=\frac{\left({\cos }^{2}A-{\sin }^{2}A\right)\left(1-{\sin }^{2}A\cdot {\cos }^{2}A\right)}{{\sin }^{4}A\cdot {\cos }^{4}A}-\left[\frac{\left({\cos }^{2}A-{\sin }^{2}A\right)-{\cos }^{2}A\cdot {\sin }^{2}A\left({\cos }^{2}A-{\sin }^{2}A\right)}{{\sin }^{4}co{s}^{4}A}\right]$ $=0$