Step 1 — Use the two Pythagorean identities:

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

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

Step 2 — Substitute:

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

Step 3 — Write in terms of $\sin$ and $\cos$:

$$= \frac{\dfrac{1}{\cos^2 A}}{\dfrac{1}{\sin^2 A}} = \frac{\sin^2 A}{\cos^2 A}$$

Step 4 — Recognise the ratio:

$$= \tan^2 A$$

$$\boxed{\text{Answer} = \tan^2 A}$$

Check with $A = 45^\circ$:

$$\frac{1 + 1}{1 + 1} = 1 = \tan^2 45^\circ \quad \checkmark$$

Note: Getting $\cot^2 A$ (option c) means the fraction was inverted by mistake.

Correct option: (d) $\tan^2 A$