Suppose $x\in \left[0,\frac{\pi }{2}\right]$  and  ${\log }_{24\sin x}^{ }\left(24\cos x\right)=\frac{3}{2}$  value of  $\cos e{c}^{2}x$  is 

$\begin{array}{c}{\log }_{24\sin x}^{ }\left(24\cos x\right)=\frac{3}{2}\\ \log 24\cos x=\frac{3}{2}\log 24\sin x\\ \sqrt{{\left(24\right)}^3{\sin }^{3}x}=24\cos x\\ {\left(24\right)}^3{\sin }^{3}x={\left(24\right)}^2{\cos }^{2}x\\ 24{\sin }^{3}x=1-{\sin }^{2}x\\ 24{\sin }^{3}x+{\sin }^{2}x-1=0\\ \sin x=\frac{1}{3}\\ cose{c}^{2}x=9\end{array}$