If `sin theta = 5//13`, then find the value of (cot`theta - tan theta)//2cot theta` ?

Options

A. 143/195
B. 119/254
C. 173/288
D. 119/288

Answer

To solve the problem, we need to find the value of \((\cot \theta - \tan \theta) / (2 \cot \theta)\) given that \(\sin \theta = \frac{5}{13}\).

### Step-by-Step Solution:

1. **Identify the values of sine, cosine, and tangent**:
- Given \(\sin \theta = \frac{5}{13}\), we can use the Pythagorean theorem to find \(\cos \theta\).
- In a right triangle, if \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\), then the opposite side is 5 and the hypotenuse is 13.
- To find the adjacent side, we use:
\[ \text{adjacent} = \sqrt{(\text{hypotenuse})^2 - (\text{opposite})^2} = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12 \]
- Therefore, \(\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12}{13}\).

2. **Calculate \(\tan \theta\) and \(\cot \theta\)**:
- \(\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{5/13}{12/13} = \frac{5}{12}\)
- \(\cot \theta = \frac{1}{\tan \theta} = \frac{12}{5}\)

3. **Substitute values into the expression**:
- We need to calculate \((\cot \theta - \tan \theta) / (2 \cot \theta)\):
- Substitute \(\cot \theta\) and \(\tan \theta\):
\[ \cot \theta - \tan \theta = \frac{12}{5} - \frac{5}{12} \]
- To subtract these fractions, we need a common denominator:
\[ \frac{12}{5} = \frac{12 \times 12}{5 \times 12} = \frac{144}{60}, \quad \frac{5}{12} = \frac{5 \times 5}{12 \times 5} = \frac{25}{60} \]
- Now, subtract:
\[ \cot \theta - \tan \theta = \frac{144}{60} - \frac{25}{60} = \frac{119}{60} \]

4. **Calculate \(2 \cot \theta\)**:
- \(2 \cot \theta = 2 \times \frac{12}{5} = \frac{24}{5}\)

5. **Final calculation**:
- Now substitute these values into the expression:
\[ \frac{\cot \theta - \tan \theta}{2 \cot \theta} = \frac{\frac{119}{60}}{\frac{24}{5}} = \frac{119}{60} \times \frac{5}{24} = \frac{119 \times 5}{60 \times 24} = \frac{595}{1440} \]
- Simplifying \(\frac{595}{1440}\) gives us the final answer.

### Final Answer:
The value of \((\cot \theta - \tan \theta) / (2 \cot \theta)\) is \(\frac{119}{288}\).