If A is in the fourth quadrant and `cosA=(5)/(13)`, find the value of `(13sinA+5secA)/(5tanA+6" cosec A")`:

Options

A. `-2//37`
B. `-3//27`
C. `2//37`
D. can't be determined

Answer

To solve the problem step by step, we will follow the trigonometric identities and properties of angles in the fourth quadrant.

### Step 1: Identify the values of sin A and sec A
Given that \( \cos A = \frac{5}{13} \), we know that in the fourth quadrant, cosine is positive and sine is negative. We can use the Pythagorean identity:

\[ \sin^2 A + \cos^2 A = 1 \]

Substituting the value of \( \cos A \):

\[ \sin^2 A + \left(\frac{5}{13}\right)^2 = 1 \]

Calculating \( \left(\frac{5}{13}\right)^2 \):

\[ \sin^2 A + \frac{25}{169} = 1 \]

Subtracting \( \frac{25}{169} \) from both sides:

\[ \sin^2 A = 1 - \frac{25}{169} = \frac{169 - 25}{169} = \frac{144}{169} \]

Taking the square root (and considering that sine is negative in the fourth quadrant):

\[ \sin A = -\frac{12}{13} \]

Now, we can find \( \sec A \):

\[ \sec A = \frac{1}{\cos A} = \frac{1}{\frac{5}{13}} = \frac{13}{5} \]

### Step 2: Substitute values into the expression
We need to evaluate:

\[ \frac{13 \sin A + 5 \sec A}{5 \tan A + 6 \csc A} \]

Substituting \( \sin A \) and \( \sec A \):

\[ = \frac{13 \left(-\frac{12}{13}\right) + 5 \left(\frac{13}{5}\right)}{5 \tan A + 6 \csc A} \]

Calculating the numerator:

\[ = \frac{-12 + 13}{5 \tan A + 6 \csc A} = \frac{1}{5 \tan A + 6 \csc A} \]

### Step 3: Find \( \tan A \) and \( \csc A \)
We already have \( \sin A \) and \( \cos A \):

\[ \tan A = \frac{\sin A}{\cos A} = \frac{-\frac{12}{13}}{\frac{5}{13}} = -\frac{12}{5} \]

And \( \csc A \):

\[ \csc A = \frac{1}{\sin A} = \frac{1}{-\frac{12}{13}} = -\frac{13}{12} \]

### Step 4: Substitute \( \tan A \) and \( \csc A \) into the denominator
Now substituting these values into the denominator:

\[ = 5 \left(-\frac{12}{5}\right) + 6 \left(-\frac{13}{12}\right) \]

Calculating the denominator:

\[ = -12 - \frac{78}{12} = -12 - 6.5 = -18.5 = -\frac{37}{2} \]

### Step 5: Final expression
Now substituting back into the expression:

\[ = \frac{1}{-\frac{37}{2}} = -\frac{2}{37} \]

Thus, the final answer is:

\[ \boxed{-\frac{2}{37}} \]