jmc

Since $\cos B=\frac{6}{10}$, and the length of the hypotenuse is $BC=10$, $AB=6$. Then, from the Pythagorean Theorem, we have $$\begin{array}{rl}A{B}^2+A{C}^2& =B{C}^2\\ \Rightarrow AC& =\sqrt{B{C}^2-A{B}^2}\\ & =\sqrt{10^2-6^2}\\ & =\sqrt{64}\\ & =8.\end{array}$$Therefore, $\tan C=\frac{AB}{AC}=\frac{6}{8}=\boxed{\frac{3}{4}}$.