Step 1: Find cos(45 degrees) and its square.
We know $\cos 45^\circ = \frac{1}{\sqrt{2}}$, so $\cos^2 45^\circ = \frac{1}{2}$.
Step 2: Find cos(135 degrees) and its square.
$135^\circ = 180^\circ - 45^\circ$ lies in the second quadrant where cosine is negative. So $\cos 135^\circ = -\cos 45^\circ = -\frac{1}{\sqrt{2}}$, and $\cos^2 135^\circ = \frac{1}{2}$. Squaring removes the negative sign.
Step 3: Find cos(225 degrees) and its square.
$225^\circ = 180^\circ + 45^\circ$ lies in the third quadrant where cosine is also negative. So $\cos 225^\circ = -\cos 45^\circ = -\frac{1}{\sqrt{2}}$, and $\cos^2 225^\circ = \frac{1}{2}$.
Step 4: Find cos(315 degrees) and its square.
$315^\circ = 360^\circ - 45^\circ$ lies in the fourth quadrant where cosine is positive. So $\cos 315^\circ = \cos 45^\circ = \frac{1}{\sqrt{2}}$, and $\cos^2 315^\circ = \frac{1}{2}$.
Step 5: Notice the pattern.
All four angles are symmetric around the axes and differ only by sign in cosine, but since we are squaring, each contributes $\frac{1}{2}$.
Step 6: Sum all four values.
$\cos^2 45^\circ + \cos^2 135^\circ + \cos^2 225^\circ + \cos^2 315^\circ = \frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2} = 2$. \[ \boxed{2} \]