$\dfrac{1}{\cos^2\alpha}$
$=\dfrac{\sin^2\alpha+\cos^2\alpha}{\cos^2\alpha}$
$=\dfrac{\sin^2\alpha}{\cos^2\alpha}+\dfrac{\cos^2\alpha}{\cos^2\alpha}$
$=\tan^2\alpha+1$
$\dfrac{1}{\sin^2\alpha}$
$=\dfrac{\sin^2\alpha+\cos^2\alpha}{\sin^2\alpha}$
$=\dfrac{\sin^2\alpha}{\sin^2\alpha}+\dfrac{\cos^2\alpha}{\sin^2\alpha}$
$=1+\cot^2\alpha$