Question

If \( \sec \theta = \frac{13}{12} \), then \( \cot \theta = \):

• A. \( \frac{5}{12} \)
• B. \( \frac{5}{13} \)
• C. \( \frac{12}{5} \)
• D. \( \frac{13}{5} \)

Hint:

Use the identity \( \sec^2 \theta = 1 + \tan^2 \theta \) to find \( \tan \theta \), and then use \( \cot \theta = \frac{1}{\tan \theta} \) to find \( \cot \theta \).

Answer 1

The Correct Option is C

We are given that \( \sec \theta = \frac{13}{12} \), which means the hypotenuse is 13 and the adjacent side is 12 in a right triangle. From the identity \( \sec^2 \theta = 1 + \tan^2 \theta \), we can find \( \tan \theta \): \[ \sec^2 \theta = \left( \frac{13}{12} \right)^2 = \frac{169}{144}. \] Thus: \[ \tan^2 \theta = \sec^2 \theta - 1 = \frac{169}{144} - 1 = \frac{169}{144} - \frac{144}{144} = \frac{25}{144}. \] So: \[ \tan \theta = \frac{5}{12}. \] Now, using the identity \( \cot \theta = \frac{1}{\tan \theta} \), we find: \[ \cot \theta = \frac{1}{\frac{5}{12}} = \frac{12}{5}. \] Therefore, \( \cot \theta = \boxed{\frac{12}{5}} \).