MY MATH PROBLEM IS
How to find the sum of all possible result of $$\frac{8\sin^2x\cos x - \sqrt{3}\sin x - \cos x}{\sin{(x + \frac{7\pi}{2})} - \sqrt{3}\cos{(x-\frac{3\pi}{2}})} = 0 \tag*{Eq.(*)} \label{Eq}$$ for $x$ in range $[0,1007\pi]$
- I just can find solution for the equation and I have no idea about the sum
$$\sin\left(x + \frac{7\pi}{2}\right) - \sqrt{3}\cos\left(x - \frac{3\pi}{2}\right) \neq 0$$ So
$x \neq \frac{\pi}{6} + k2\pi $ and $x \neq \frac{5\pi}{6} + k2\pi$
while $k$ is any integer
$$\ref{Eq} \iff 4\sin{x}\sin{2x} - \sqrt{3}\sin{x} - \cos{x} = 0$$ $$\iff -2(\cos{3x} - \cos{x}) - \sqrt{3}\sin{x} - \cos{x} = 0$$ $$\iff \sin\left(\frac{-\pi}{2} + 3x\right) = \sin\left(x - \frac{\pi}{6}\right)$$ $x = \frac{\pi}{6} + k2\pi $ (Removed) or $x = \frac{5\pi}{12} + k2\pi$ (Taken)
We have $$0\le\frac{5\pi}{12} + k2\pi\le1007\pi$$ $$\frac{-5}{24}\le k \le \frac{12079}{24}$$ $$k \in [0,503]$$
Till here, i don't know how to find the sum. Please give me some HINT for this problem