1. **State the problem:** We are given the function $f(x) = (1 + \tan^2 x) \tan^2 x$ and need to simplify it. 2. **Recall the trigonometric identity:** One important identity is $1 + \tan^2 x = \sec^2 x$. 3. **Apply the identity:** Substitute $1 + \tan^2 x$ with $\sec^2 x$ in the function: $$f(x) = \sec^2 x \cdot \tan^2 x$$ 4. **Express in terms of sine and cosine:** Recall that $\tan x = \frac{\sin x}{\cos x}$ and $\sec x = \frac{1}{\cos x}$, so $$f(x) = \left(\frac{1}{\cos^2 x}\right) \left(\frac{\sin^2 x}{\cos^2 x}\right) = \frac{\sin^2 x}{\cos^4 x}$$ 5. **Final simplified form:** The function simplifies to $$f(x) = \frac{\sin^2 x}{\cos^4 x}$$ This is the simplest form using basic trigonometric functions.