jmc

Since $\angle Z$ is the right angle, our triangle is shown below:



Since $\cos Y=\frac{7}{25}$, we have $\cos Y=\frac{YZ}{XY}=\frac{YZ}{25}=\frac{7}{25}$. From this, we can see that $YZ=25\cdot \frac{7}{25}=7$. Then, from the Pythagorean Theorem, $XZ=\sqrt{X{Y}^2-Y{Z}^2}=\sqrt{25^2-7^2}=\sqrt{576}=24$. Finally, we have $\tan X=\frac{YZ}{XZ}=\boxed{\frac{7}{24}}$.