Step 1: Simplify the LHS using $1+\cot^{2}A=\operatorname{cosec}^{2}A$ and $1+\tan^{2}A=\sec^{2}A$.
$$\text{LHS}=\frac{\operatorname{cosec}^{2}A}{\sec^{2}A}=\frac{1/\sin^{2}A}{1/\cos^{2}A}=\frac{\cos^{2}A}{\sin^{2}A}=\cot^{2}A$$
Step 2: Now take the RHS and write everything in terms of $\sin A$ and $\cos A$.
$$\text{RHS}=\left(\frac{1-\dfrac{\cos A}{\sin A}}{1-\dfrac{\sin A}{\cos A}}\right)^{2}$$
Step 3: Simplify the numerator and denominator separately.
$$=\left(\frac{\dfrac{\sin A-\cos A}{\sin A}}{\dfrac{\cos A-\sin A}{\cos A}}\right)^{2}$$
Step 4: Divide the fractions.
$$=\left(\frac{(\sin A-\cos A)\cos A}{\sin A(\cos A-\sin A)}\right)^{2}=\left(\frac{-\cos A}{\sin A}\right)^{2}$$
Step 5: Square it (the negative sign disappears).
$$=\frac{\cos^{2}A}{\sin^{2}A}=\cot^{2}A$$
Step 6: LHS $=$ RHS $=\cot^{2}A$. Hence proved.