If `cosA=(5)/(13)` then what is the value of `(sinA-cotA)/(2tanA)` ?

Options

A. `(395)/(3644)`
B. `(395)/(3844)`
C. `(395)/(3744)`
D. `(385)/(3744)`

Answer

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

**Given:**
\[ \cos A = \frac{5}{13} \]

We need to find the value of:
\[ \frac{\sin A - \cot A}{2 \tan A} \]

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

Using the Pythagorean identity:
\[ \sin^2 A + \cos^2 A = 1 \]
We can substitute \(\cos A\):
\[ \sin^2 A + \left(\frac{5}{13}\right)^2 = 1 \]
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:
\[ \sin A = \frac{12}{13} \]

### Step 2: Find \(\tan A\)

Using the definition of tangent:
\[ \tan A = \frac{\sin A}{\cos A} = \frac{\frac{12}{13}}{\frac{5}{13}} = \frac{12}{5} \]

### Step 3: Find \(\cot A\)

Using the definition of cotangent:
\[ \cot A = \frac{1}{\tan A} = \frac{5}{12} \]

### Step 4: Substitute values into the expression

Now we substitute \(\sin A\), \(\cot A\), and \(\tan A\) into the expression:
\[ \frac{\sin A - \cot A}{2 \tan A} = \frac{\frac{12}{13} - \frac{5}{12}}{2 \cdot \frac{12}{5}} \]

### Step 5: Simplify the numerator

Finding a common denominator for the numerator:
\[ \frac{12}{13} - \frac{5}{12} = \frac{12 \cdot 12 - 5 \cdot 13}{13 \cdot 12} = \frac{144 - 65}{156} = \frac{79}{156} \]

### Step 6: Simplify the denominator

Calculating the denominator:
\[ 2 \tan A = 2 \cdot \frac{12}{5} = \frac{24}{5} \]

### Step 7: Combine the results

Now we can combine the results:
\[ \frac{\frac{79}{156}}{\frac{24}{5}} = \frac{79}{156} \cdot \frac{5}{24} = \frac{79 \cdot 5}{156 \cdot 24} \]

### Step 8: Calculate the final value

Calculating:
\[ \frac{395}{3744} \]

Thus, the final answer is:
\[ \frac{395}{3744} \]