To find the values of sec C and tan C given that \( \cos C = \frac{5}{9} \) and \( \cot C = \frac{5}{\sqrt{56}} \), we can use trigonometric identities.
Step 1: Find \( \sec C \)
The secant function is the reciprocal of the cosine function:
\[ \sec C = \frac{1}{\cos C} \]
Substituting the value of \( \cos C \):
\[ \sec C = \frac{1}{\frac{5}{9}} = \frac{9}{5} \]
Step 2: Find \( \tan C \)
The cotangent function gives us \( \tan C \) as the reciprocal of \( \cot C \):
\[ \tan C = \frac{1}{\cot C} \]
Substituting the value of \( \cot C \):
\[ \tan C = \frac{1}{\frac{5}{\sqrt{56}}} = \frac{\sqrt{56}}{5} \]
Summary of Results
- \( \sec C = \frac{9}{5} \)
- \( \tan C = \frac{\sqrt{56}}{5} \)
Finally, since \( \sqrt{56} = \sqrt{4 \cdot 14} = 2\sqrt{14} \), we can express \( \tan C \) more neatly:
\[ \tan C = \frac{2\sqrt{14}}{5} \]
Thus, the final answers are:
- \( \sec C = \frac{9}{5} \)
- \( \tan C = \frac{2\sqrt{14}}{5} \)