Step 1: Spot the pattern.
We are asked for $\sin^2 15^\circ+\cos^2 15^\circ$. This has the shape $\sin^2\theta+\cos^2\theta$, which is a famous identity.
Step 2: State the identity.
For every angle $\theta$, \[ \sin^2\theta+\cos^2\theta=1. \] This is the Pythagorean identity and it never depends on the size of the angle.
Step 3: Set the angle.
Here $\theta=15^\circ$, a single fixed angle, so the identity applies directly.
Step 4: Substitute.
\[ \sin^2 15^\circ+\cos^2 15^\circ=1. \]
Step 5: Note no calculation is needed.
We do not need the actual values of $\sin 15^\circ$ or $\cos 15^\circ$; the identity gives the result at once.
Step 6: Choose the option.
The value is $1$, which is option 3.
\[ \boxed{1} \]