Square Roots Give Two Signs

$2{\cos\left(x\right)}^{2} = 1$ has no cos x term, so take the square root instead of factoring: ${\cos\left(x\right)}^{2} = \frac{1}{2}$, so cos x = √2/2 or cos x = −√2/2. Both signs count.

The positive case gives 45° and 315°, the negative case 135° and 225°: four answers, one in each quadrant. Keeping only the positive root halves the answer, the trap of x² = 4 with only x = 2.