Answer: 192

Small Hint:

Rewrite the equation in exponential form and square both sides to get $\cos^2 x = 24 \sin^3 x$

Big Hint:

Replace $\cos^2 x$ by $1 - \sin^2 x$ and look for a rational root of the resulting cubic in $\sin x$

Solution:

In exponential form the equation says $(24 \sin x)^{\frac{3}{2}} = 24 \cos x.$ Squaring gives $24^3 \sin^3 x = 24^2 \cos^2 x,$ so $\cos^2 x = 24 \sin^3 x.$

Writing $s = \sin x$ and using $\cos^2 x = 1 - s^2,$ we get $24s^3 + s^2 - 1 = 0,$ which factors as $(3s - 1)(8s^2 + 3s + 1) = 0.$ The quadratic factor has negative discriminant, so $\sin x = \frac{1}{3}.$

Then $$\begin{aligned} 24 \cot^2 x &= 24 \cdot \frac{1 - \sin^2 x}{\sin^2 x} \\ &= 24 \cdot \frac{\frac{8}{9}}{\frac{1}{9}} \\ &= 24 \cdot 8 = 192. \end{aligned}$$