If `cosA = 5/13`, find the value of tan A + cot A

Options

A. `169/60`
B. `12/13`
C. 1
D. `60/169`

Answer

To solve the problem, we need to find the value of \( \tan A + \cot A \) given that \( \cos A = \frac{5}{13} \).

### Step 1: Write the expression for \( \tan A + \cot A \)

We know that:
\[ \tan A = \frac{\sin A}{\cos A} \quad \text{and} \quad \cot A = \frac{\cos A}{\sin A} \]
Thus, we can write:
\[ \tan A + \cot A = \frac{\sin A}{\cos A} + \frac{\cos A}{\sin A} \]

### Step 2: Combine the fractions

To combine the fractions, we find a common denominator, which is \( \sin A \cos A \):
\[ \tan A + \cot A = \frac{\sin^2 A + \cos^2 A}{\sin A \cos A} \]

### Step 3: Use the Pythagorean identity

From the Pythagorean identity, we know that:
\[ \sin^2 A + \cos^2 A = 1 \]
So we can substitute this into our expression:
\[ \tan A + \cot A = \frac{1}{\sin A \cos A} \]

### Step 4: Find \( \sin A \)

We know \( \cos A = \frac{5}{13} \). To find \( \sin A \), we use the identity:
\[ \sin^2 A + \cos^2 A = 1 \implies \sin^2 A = 1 - \cos^2 A \]
Calculating \( \cos^2 A \):
\[ \cos^2 A = \left(\frac{5}{13}\right)^2 = \frac{25}{169} \]
Thus,
\[ \sin^2 A = 1 - \frac{25}{169} = \frac{169 - 25}{169} = \frac{144}{169} \]
Taking the square root:
\[ \sin A = \sqrt{\frac{144}{169}} = \frac{12}{13} \]

### Step 5: Substitute \( \sin A \) and \( \cos A \) into the expression

Now we substitute \( \sin A \) and \( \cos A \) back into the expression for \( \tan A + \cot A \):
\[ \tan A + \cot A = \frac{1}{\sin A \cos A} = \frac{1}{\left(\frac{12}{13}\right) \left(\frac{5}{13}\right)} \]

### Step 6: Calculate \( \sin A \cos A \)

Calculating \( \sin A \cos A \):
\[ \sin A \cos A = \frac{12}{13} \cdot \frac{5}{13} = \frac{60}{169} \]
Thus,
\[ \tan A + \cot A = \frac{1}{\frac{60}{169}} = \frac{169}{60} \]

### Final Answer

The value of \( \tan A + \cot A \) is:
\[ \boxed{\frac{169}{60}} \]