We know that,

$\cos A=\frac{1}{\sec A}$

Also, $\sin ^{2} A+\cos ^{2} A=1$

$\sin ^{2} A=1-\cos ^{2} A$

$\sin A=\sqrt{1-\left(\frac{1}{\sec A}\right)^{2}}$

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

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

$\tan ^{2} A=\sec ^{2} A-1$

$\tan A =\sqrt{\sec ^{2} A -1}$

$\cot A =\frac{\cos A }{\sin A } =\frac{\frac{1}{\sec A}}{\frac{\sqrt{\sec ^{2} A-1}}{\sec A}}$

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

$\operatorname{cosec} A =\frac{1}{\sin A }=\frac{\sec A }{\sqrt{\sec ^{2} A -1}}$