Step 1: Write down what we must simplify.
We need to evaluate $1+\sec^2x\,\sin^2x$ and match it with one of the options.
Step 2: Express the secant in basic terms.
Recall $\sec x=\dfrac{1}{\cos x}$, so $\sec^2x=\dfrac{1}{\cos^2x}$.
Step 3: Simplify the product term.
Then \[ \sec^2x\,\sin^2x=\frac{\sin^2x}{\cos^2x}=\tan^2x. \] Step 4: Rewrite the whole expression.
The expression becomes $1+\tan^2x$.
Step 5: Use the Pythagorean identity.
We know the identity $1+\tan^2x=\sec^2x$. So the expression simplifies straight to $\sec^2x$.
Step 6: State the final answer.
Hence $1+\sec^2x\,\sin^2x=\sec^2x$.
\[ \boxed{\sec^2x} \]