Step 1: Understanding the Concept:
We use the trigonometric identity $1 = \sin^2 x + \cos^2 x$ to break the product in the integrand into a sum that is easier to integrate.
Step 2: Formula Derivation:
Substitute $\sec^2 x = \frac{1}{\cos^2 x}$ and $\csc^2 x = \frac{1}{\sin^2 x}$:
$$\int \frac{1}{\cos^2 x \cdot \sin^2 x} \, dx = \int \frac{\sin^2 x + \cos^2 x}{\sin^2 x \cos^2 x} \, dx$$
$$\int \left( \frac{\sin^2 x}{\sin^2 x \cos^2 x} + \frac{\cos^2 x}{\sin^2 x \cos^2 x} \right) \, dx = \int (\sec^2 x + \csc^2 x) \, dx$$
Step 3: Explanation:
The integral of $\sec^2 x$ is $\tan x$, and the integral of $\csc^2 x$ is $-\cot x$.
$$\int \sec^2 x \, dx + \int \csc^2 x \, dx = \tan x - \cot x + C$$
Step 4: Final Answer:
The correct option is (c).