If `cos alpha=(5)/(13)` and `alpha` lies in the fourth quadrant, find the value of `(2-3cot alpha)/(4-9sqrt(sec^(2)alpha-1))`.

Answer

To solve the problem, we start with the given information:

**Given:**
- \( \cos \alpha = \frac{5}{13} \)
- \( \alpha \) is in the fourth quadrant.

### Step 1: Find \( \sin \alpha \)

Using the Pythagorean identity:
\[ \sin^2 \alpha + \cos^2 \alpha = 1 \]
Substituting the value of \( \cos \alpha \):
\[ \sin^2 \alpha + \left(\frac{5}{13}\right)^2 = 1 \]
\[ \sin^2 \alpha + \frac{25}{169} = 1 \]
\[ \sin^2 \alpha = 1 - \frac{25}{169} = \frac{169 - 25}{169} = \frac{144}{169} \]
Taking the square root:
\[ \sin \alpha = -\sqrt{\frac{144}{169}} = -\frac{12}{13} \]
(Note: \( \sin \alpha \) is negative in the fourth quadrant.)

### Step 2: Find \( \cot \alpha \)

Using the definition of cotangent:
\[ \cot \alpha = \frac{\cos \alpha}{\sin \alpha} = \frac{\frac{5}{13}}{-\frac{12}{13}} = -\frac{5}{12} \]

### Step 3: Find \( \sec^2 \alpha \)

Using the identity:
\[ \sec^2 \alpha = 1 + \tan^2 \alpha \]
First, find \( \tan \alpha \):
\[ \tan \alpha = \frac{\sin \alpha}{\cos \alpha} = \frac{-\frac{12}{13}}{\frac{5}{13}} = -\frac{12}{5} \]
Now, calculate \( \tan^2 \alpha \):
\[ \tan^2 \alpha = \left(-\frac{12}{5}\right)^2 = \frac{144}{25} \]
Thus,
\[ \sec^2 \alpha = 1 + \frac{144}{25} = \frac{25}{25} + \frac{144}{25} = \frac{169}{25} \]

### Step 4: Calculate \( \sec^2 \alpha - 1 \)

\[ \sec^2 \alpha - 1 = \frac{169}{25} - 1 = \frac{169}{25} - \frac{25}{25} = \frac{144}{25} \]

### Step 5: Calculate \( \sqrt{\sec^2 \alpha - 1} \)

\[ \sqrt{\sec^2 \alpha - 1} = \sqrt{\frac{144}{25}} = \frac{12}{5} \]

### Step 6: Substitute into the expression

Now substitute \( \cot \alpha \) and \( \sqrt{\sec^2 \alpha - 1} \) into the expression:
\[ \frac{2 - 3 \cot \alpha}{4 - 9 \sqrt{\sec^2 \alpha - 1}} = \frac{2 - 3 \left(-\frac{5}{12}\right)}{4 - 9 \left(\frac{12}{5}\right)} \]

### Step 7: Simplify the numerator

\[ 2 - 3 \left(-\frac{5}{12}\right) = 2 + \frac{15}{12} = 2 + \frac{5}{4} = \frac{8}{4} + \frac{5}{4} = \frac{13}{4} \]

### Step 8: Simplify the denominator

\[ 4 - 9 \left(\frac{12}{5}\right) = 4 - \frac{108}{5} = \frac{20}{5} - \frac{108}{5} = -\frac{88}{5} \]

### Step 9: Final expression

Now we can combine the results:
\[ \frac{\frac{13}{4}}{-\frac{88}{5}} = \frac{13 \times 5}{4 \times -88} = \frac{65}{-352} = -\frac{65}{352} \]

Thus, the final answer is:
\[ \boxed{-\frac{65}{352}} \]