cos(4v) + cos(v) = 0 I am given the following equation: 
$$\cos 4v + \cos v = 0$$
My attempt: 
$$\cos4v = -\cos v$$
$$\cos4v = \cos(\pi \pm v)$$
$$4v = \pm \pi \pm v + 2\pi n$$

$$4v_1 = \pi + v_1 + 2\pi n$$
$$4v_2 = -\pi - v_2 + 2\pi n$$
$$4v_3 = +\pi - v_3 + 2\pi n$$
$$4v_4 = -\pi + v_4 + 2\pi n$$

$$v_1 = \frac{\pi}{3} + \frac{2\pi n}{3}$$
$$v_2 = -\frac{\pi}{5} + \frac{2\pi n}{5}$$
$$v_3 = \frac{\pi}{5} + \frac{2\pi n}{5}$$
$$v_4 = -\frac{\pi}{3} + \frac{2\pi n}{3}$$
However, the answer is simply the positive solutions i.e: 
$$v_1 = \frac{\pi}{3} + \frac{2\pi n}{3}$$
$$v_3 = \frac{\pi}{5} + \frac{2\pi n}{5}$$
Why? What am I missing?

HINT: use that $$\cos(x)-\cos(y)=-2\sin\left(\frac{x-y}{2}\right)\sin\left(\frac{x+y}{2}\right)$$
sorry i misreaded the post use that
$$\cos(x)+\cos(y)=2 \cos \left(\frac{x}{2}-\frac{y}{2}\right) \cos
   \left(\frac{x}{2}+\frac{y}{2}\right)$$
