We know that,

$\operatorname{cosec}^{2} A=1+\cot ^{2} A$

$\frac{1}{\operatorname{cosec}^{2} A}=\frac{1}{1+\cot ^{2} A}$

$\sin ^{2} A=\frac{1}{1+\cot ^{2} A}$

$\sin A=\pm \frac{1}{\sqrt{1+\cot ^{2} A}}$

$\sqrt{1+\cot ^{2} A}$ will always be positive as we are adding two positive quantities.

Therefore, $\sin A =\frac{1}{\sqrt{1+\cot ^{2} A }}$

We know that, $\tan A =\frac{\sin A }{\cos A }$

However, $\cot A=\frac{\cos A}{\sin A}$

Therefore, $\tan A =\frac{1}{\cot A }$

Also, $\sec ^{2} A=1+\tan ^{2} A$

$=1+\frac{1}{\cot ^{2} A}$

$=\frac{\cot ^{2} A+1}{\cot ^{2} A}$

$\sec A=\frac{\sqrt{\cot ^{2} A+1}}{\cot A}$