Bài giải:

$\cos \alpha =\frac{3}{4}\Rightarrow {{\cos }^{2}}\alpha =\frac{9}{16}$

${{\sin }^{2}}\alpha +{{\cos }^{2}}\alpha =1\Leftrightarrow {{\sin }^{2}}\alpha +\frac{9}{16}=1$

$\Rightarrow {{\sin }^{2}}\alpha =\frac{7}{16}$

${{0}^{o}}<\alpha <{{90}^{o}}\Rightarrow \sin \alpha >0\Rightarrow \sin \alpha =\frac{\sqrt{7}}{4}$

$\cot \alpha =\frac{\cos \alpha }{\sin \alpha }=\frac{3}{4}:\frac{\sqrt{7}}{4}=\frac{3}{\sqrt{7}}$