1. The problem is to understand why $\cos(\pi + a)$ equals $-\cos(a)$.\n\n2. Recall the cosine addition formula: $$\cos(x + y) = \cos x \cos y - \sin x \sin y.$$\n\n3. Apply this formula with $x = \pi$ and $y = a$: $$\cos(\pi + a) = \cos \pi \cos a - \sin \pi \sin a.$$\n\n4. Substitute the known values: $\cos \pi = -1$ and $\sin \pi = 0$. So, $$\cos(\pi + a) = (-1) \cdot \cos a - 0 \cdot \sin a = -\cos a.$$\n\n5. Therefore, $\cos(\pi + a) = -\cos(a)$ because adding $\pi$ to the angle flips the cosine value's sign.