\begin{align*} &\sin^2 10^\circ + \cos^2 40^\circ + \sin 10^\circ \cos 40^\circ\\ =\quad&\frac{1-\cos 20^\circ}{2}+\frac{1+\cos 80^\circ}{2} + \frac{1}{2}\cdot(\sin 50^\circ -\sin 30^\circ)\\ =\quad&1 -\frac{1}{2}\cdot \sin 30^\circ+ \frac{1}{2}\cdot(-\cos 20^\circ +\cos 80^\circ + \sin 50^\circ)\\ =\quad&\frac{3}{4}+\frac{1}{2}(-2\sin 50^\circ \sin 30^\circ +\sin 50^\circ)\\ =\quad&\boxed{\frac{3}{4}} \end{align*}