Finding exact value(s) of x in $4\sin(x)\cos(x)-3\sin(x)=0$ One solution is $x = n\pi, n\in \mathbb{Z}$
The second one is $\cos(x) = \frac{3}{4}$
I do not know how to find an exact value for x in the second one.
$$\cos x=\frac34$$ naturally has the solution

$$x=\arccos\frac34.$$

As the cosine is an even function ($\cos x=\cos(-x)$) and has period $2\pi$, the solutions are

$$x=\pm\arccos\frac34+2k\pi.$$
so here i attach the graph of $y = \sin(x) \cos(x) - 3\cos(x)$ in the interval $[-4\pi,4\pi]$. There are 17 roots (intersection of $ y = 0 $ with $y = \sin(x) \cos(x) - 3\cos(x)$ ) in the interval  $[-4\pi,4\pi]$.

As you have said, you have roots of the form $n\pi$ (which are $-4\pi \ldots 4\pi)$. Those are 9 roots. 

The other 8 are in the form of $\cos(x) = \frac{3}{4}$ also as you said. Also as @Mundron Schmidt suggests, the solutions of $\cos(x) = \frac{3}{4}$ are in the form of 
\begin{equation}
x = \pm \arccos(\frac{3}{4}) + 2n \pi \qquad n \in \mathbb{Z}
  \end{equation}
which is the simplest form. And those are the 8 other roots in $[-4\pi,4\pi]$