To find the values of sec \( C \) and \( \tan C \), we start with the given values \( \cos C = \frac{5}{9} \) and \( \cot C = \frac{5}{\sqrt{56}} \).
Step 1: Finding \( \sin C \) from \( \cos C \)
We use the Pythagorean identity:
\[ \sin^2 C + \cos^2 C = 1 \]
Substituting \( \cos C \):
\[ \sin^2 C + \left( \frac{5}{9} \right)^2 = 1 \]
Calculating \( \left( \frac{5}{9} \right)^2 \):
\[ \sin^2 C + \frac{25}{81} = 1 \]
Now, we isolate \( \sin^2 C \):
\[ \sin^2 C = 1 - \frac{25}{81} = \frac{81}{81} - \frac{25}{81} = \frac{56}{81} \]
Thus, we take the positive root (since \( C \) is an acute angle):
\[ \sin C = \sqrt{\sin^2 C} = \sqrt{\frac{56}{81}} = \frac{\sqrt{56}}{9} \]
Step 2: Finding \( \sec C \)
Now we find \( \sec C \), which is the reciprocal of \( \cos C \):
\[ \sec C = \frac{1}{\cos C} = \frac{1}{\frac{5}{9}} = \frac{9}{5} \]
Step 3: Finding \( \tan C \)
We can find \( \tan C \) using the definitions of sine and cosine:
\[ \tan C = \frac{\sin C}{\cos C} \]
Substituting the values we found:
\[ \tan C = \frac{\frac{\sqrt{56}}{9}}{\frac{5}{9}} = \frac{\sqrt{56}}{5} \]
Summary of Results
The values we have calculated are:
\[ \sec C = \frac{9}{5} \] \[ \tan C = \frac{\sqrt{56}}{5} \]
Both values are boxed for clarity:
\[ \sec C = \frac{9}{5}, \quad \tan C = \frac{\sqrt{56}}{5} \]