The ascending order of $A=\cos \left({\sin }^{-1}\frac{12}{13}\right),B=\sin \left(2{\tan }^{-1}\frac{3}{4}\right),C=\tan \left(\frac{1}{2}{\cos }^{-1}\frac{1}{3}\right)$  is

$B=\sin \left(2{\tan }^{-1}\frac{3}{4}\right)=\sin \left(2{\tan }^{-1}\frac{3}{4}\right)=\sin \left({\sin }^{-1}\left(\frac{2\left(\frac{3}{4}\right)}{1+{\left(\frac{3}{4}\right)}^2}\right)\right)=\frac{24}{25}$

$C=\tan \left(\frac{1}{2}{\cos }^{-1}\frac{1}{3}\right)$

Let $\theta =\frac{1}{2}{\cos }^{-1}\frac{1}{3}\Rightarrow \cos 2\theta =\frac{1}{3}$

$\tan \theta =\sqrt{\frac{1-\cos 2\theta }{1+\cos 2\theta }}=\frac{1}{\sqrt{2}}\Rightarrow \theta ={\tan }^{-1}\left(\frac{1}{\sqrt{2}}\right)$

$C=\tan \left(\frac{1}{2}{\cos }^{-1}\frac{1}{3}\right)=\tan \left({\tan }^{-1}\left(\frac{1}{\sqrt{2}}\right)\right)=\frac{1}{\sqrt{2}}$

Ascending order     $A,C,B$