If `sec alpha=13/5, (0 lt alpha lt pi/2)`, then the value of `(2-3cotalpha)/(4-9sqrt(sec^(2)alpha-1))` is

Options

A. `-15/352`
B. `15/352`
C. `30/352`
D. `5/352`

Answer

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

**Given:**
\[ \sec \alpha = \frac{13}{5}, \quad (0 < \alpha < \frac{\pi}{2}) \]

We need to find the value of:
\[ \frac{2 - 3 \cot \alpha}{4 - 9 \sqrt{\sec^2 \alpha - 1}} \]

### Step 1: Calculate \(\sec^2 \alpha\)
We know that:
\[ \sec^2 \alpha = \frac{13^2}{5^2} = \frac{169}{25} \]

### Step 2: Calculate \(\tan^2 \alpha\)
Using the identity \(\sec^2 \alpha = 1 + \tan^2 \alpha\):
\[ \tan^2 \alpha = \sec^2 \alpha - 1 = \frac{169}{25} - 1 = \frac{169}{25} - \frac{25}{25} = \frac{144}{25} \]

### Step 3: Calculate \(\tan \alpha\)
Taking the square root:
\[ \tan \alpha = \sqrt{\frac{144}{25}} = \frac{12}{5} \]

### Step 4: Calculate \(\cot \alpha\)
Since \(\cot \alpha = \frac{1}{\tan \alpha}\):
\[ \cot \alpha = \frac{5}{12} \]

### Step 5: Calculate \(\sqrt{\sec^2 \alpha - 1}\)
We already calculated \(\tan^2 \alpha\) which is equal to \(\sec^2 \alpha - 1\):
\[ \sqrt{\sec^2 \alpha - 1} = \sqrt{\tan^2 \alpha} = \tan \alpha = \frac{12}{5} \]

### Step 6: Substitute values into the expression
Now we substitute \(\cot \alpha\) and \(\sqrt{\sec^2 \alpha - 1}\) into the expression:
\[ \frac{2 - 3 \cot \alpha}{4 - 9 \sqrt{\sec^2 \alpha - 1}} = \frac{2 - 3 \cdot \frac{5}{12}}{4 - 9 \cdot \frac{12}{5}} \]

### Step 7: Simplify the numerator
Calculating the numerator:
\[ 2 - 3 \cdot \frac{5}{12} = 2 - \frac{15}{12} = \frac{24}{12} - \frac{15}{12} = \frac{9}{12} = \frac{3}{4} \]

### Step 8: Simplify the denominator
Calculating the denominator:
\[ 4 - 9 \cdot \frac{12}{5} = 4 - \frac{108}{5} = \frac{20}{5} - \frac{108}{5} = \frac{20 - 108}{5} = \frac{-88}{5} \]

### Step 9: Combine the results
Now we can combine the results:
\[ \frac{\frac{3}{4}}{\frac{-88}{5}} = \frac{3}{4} \cdot \frac{5}{-88} = \frac{15}{-352} = -\frac{15}{352} \]

### Final Result
Thus, the value of the expression is:
\[ -\frac{15}{352} \]