Draw a right-angled triangle with angles $90^\circ$, $(45+x)^\circ$ and $(45-x)^\circ$.

What does Pythagoras' Theorem tell you about these angles?

Use this information to find $\sin^2 1^\circ + \sin^2 2^\circ + \, \cdots \,+ \sin^2 359 ^\circ + \sin^2 360^\circ$.