Although no exact analytical solutions for this mixed log-trigonometric equation are available, really good analytic approximations can still be derived.
Rewrite the equation
$$\frac{\cos x}{x} + \ln x\sin x = 0 $$
equivalently as,
$$ \tan x =-\frac{1}{x\ln x}$$
Observe that the rhs quickly becomes small as $x$ moves out. Since $\tan(x)$ assumes small values around $k\pi$, there would be infinite number of roots, all around $k\pi$.
To proceed, let $x=k\pi +y$ and approximate $\tan(x)$ around $k\pi$ as
$$\tan(x)=\tan(k\pi+y) \approx y \tag{1}$$
and, similarly, approximate $-1/(x\ln x)$ around $k\pi$ as
$$-\frac{1}{x\ln x} \approx
-\frac{1}{k\pi\ln (k\pi)}
+\frac{\ln(k\pi)+ 1}{[k\pi \ln (k\pi)]^2}
y\tag{2} $$
As a result, $y$ can be solved from (1) and (2),
$$y_k =
-\frac{k\pi\ln (k\pi)}
{[k\pi \ln (k\pi)]^2-\ln(k\pi) - 1}$$
And, hence, the solutions to the original equation $ x_k = k\pi + y_k$,
$$ x_k = k\pi \left[1-\frac{\ln (k\pi)}
{(k\pi)^2\ln^2 (k\pi)-\ln(k\pi) - 1} \right]\tag{3}$$
with $k=1,2,3, ... \infty$.
For illustration, the first few roots are
$$x_1 \approx \pi - 0.33334 = 2.80825 \space (2.80984)$$
$$x_2 \approx 2\pi - 0.08848= 6.19471 \space (6.19490)$$
$$x_3 \approx 3\pi - 0.04764 =9.37714 \space (9.37717)$$
$$...$$
$$x_n=n\pi$$
where, for comparison, the exact roots are provided in the parentheses.
The algebraic solutions (3) are fairly accurate, even for the very first root. The successive roots quickly approaches $n\pi$.
