Write all the other trigonometric ratios of $\angle A$ in terms of $\sec A$.

We know that,$\cos A = \frac{1}{\sec A}$.
Using the identity $\sin^2 A + \cos^2 A = 1$,we have $\sin^2 A = 1 - \cos^2 A = 1 - \frac{1}{\sec^2 A} = \frac{\sec^2 A - 1}{\sec^2 A}$.
Thus,$\sin A = \frac{\sqrt{\sec^2 A - 1}}{\sec A}$.
Using the identity $\tan^2 A + 1 = \sec^2 A$,we have $\tan^2 A = \sec^2 A - 1$.
Thus,$\tan A = \sqrt{\sec^2 A - 1}$.
Since $\cot A = \frac{1}{\tan A}$,we have $\cot A = \frac{1}{\sqrt{\sec^2 A - 1}}$.
Since $\operatorname{cosec} A = \frac{1}{\sin A}$,we have $\operatorname{cosec} A = \frac{\sec A}{\sqrt{\sec^2 A - 1}}$.