Let $\theta$ be an angle in quadrant I such that $\cos \theta=\frac{5}{13}$. Find the exact values of $\csc \theta$ and $\tan \theta$.
\[
\begin{array}{l}
\csc \theta= \\
\tan \theta=
\end{array}
\]

Step 1 ：Let \(\theta\) be an angle in quadrant I such that \(\cos \theta=\frac{5}{13}\). We are asked to find the exact values of \(\csc \theta\) and \(\tan \theta\).

Step 2 ：The cosine of an angle in the unit circle is defined as the x-coordinate of the point where the terminal side of the angle intersects the unit circle. The cosecant of an angle is the reciprocal of the sine of the angle. Since sine is the y-coordinate of the point where the terminal side of the angle intersects the unit circle, we can use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to find the sine of the angle, and then take the reciprocal to find the cosecant.

Step 3 ：The tangent of an angle is defined as the sine of the angle divided by the cosine of the angle. Once we have the sine and cosine, we can easily find the tangent.

Step 4 ：Using the given \(\cos \theta = \frac{5}{13}\), we can find \(\sin \theta = \sqrt{1 - \cos^2 \theta} = \sqrt{1 - \left(\frac{5}{13}\right)^2} = \frac{12}{13}\).

Step 5 ：Then, we can find \(\csc \theta = \frac{1}{\sin \theta} = \frac{13}{12}\) and \(\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{12}{5}\).

Step 6 ：So, the exact values of \(\csc \theta\) and \(\tan \theta\) are \(\boxed{\frac{13}{12}}\) and \(\boxed{\frac{12}{5}}\) respectively.