1. **State the problem:** Find the derivative of the function $f(x) = \sin^2(x) \times \tan(x)$. 2. **Recall the formula:** To differentiate a product of two functions $u(x)$ and $v(x)$, use the product rule: $$\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$$ 3. **Identify the functions:** Let $u(x) = \sin^2(x)$ and $v(x) = \tan(x)$. 4. **Differentiate $u(x)$:** Use the chain rule: $$u'(x) = 2\sin(x) \cdot \cos(x)$$ 5. **Differentiate $v(x)$:** $$v'(x) = \sec^2(x)$$ 6. **Apply the product rule:** $$f'(x) = u'(x)v(x) + u(x)v'(x) = 2\sin(x)\cos(x) \cdot \tan(x) + \sin^2(x) \cdot \sec^2(x)$$ 7. **Final answer:** $$\boxed{f'(x) = 2\sin(x)\cos(x)\tan(x) + \sin^2(x)\sec^2(x)}$$