If ` cos alpha = (12)/(13)` then what is the value of ` ( cosec alpha )/( 1 + sec alpha )` ?

Options

A. `(156)/(125)`
B. `(125)/(156)`
C. `(5)/(7)`
D. None of these

Answer

To solve the problem, we need to find the value of \( \frac{\csc \alpha}{1 + \sec \alpha} \) given that \( \cos \alpha = \frac{12}{13} \).

### Step-by-step Solution:

1. **Identify the values of sine and cosine:**
We know that \( \cos \alpha = \frac{12}{13} \). To find \( \sin \alpha \), we can use the Pythagorean identity:
\[ \sin^2 \alpha + \cos^2 \alpha = 1 \]
Substituting the value of \( \cos \alpha \):
\[ \sin^2 \alpha + \left(\frac{12}{13}\right)^2 = 1 \]
\[ \sin^2 \alpha + \frac{144}{169} = 1 \]
\[ \sin^2 \alpha = 1 - \frac{144}{169} = \frac{169 - 144}{169} = \frac{25}{169} \]
Therefore,
\[ \sin \alpha = \frac{5}{13} \]

2. **Calculate cosecant and secant:**
The cosecant and secant functions are defined as:
\[ \csc \alpha = \frac{1}{\sin \alpha} = \frac{1}{\frac{5}{13}} = \frac{13}{5} \]
\[ \sec \alpha = \frac{1}{\cos \alpha} = \frac{1}{\frac{12}{13}} = \frac{13}{12} \]

3. **Substituting into the expression:**
Now we substitute \( \csc \alpha \) and \( \sec \alpha \) into the expression \( \frac{\csc \alpha}{1 + \sec \alpha} \):
\[ \frac{\csc \alpha}{1 + \sec \alpha} = \frac{\frac{13}{5}}{1 + \frac{13}{12}} \]

4. **Simplifying the denominator:**
To simplify the denominator:
\[ 1 + \frac{13}{12} = \frac{12}{12} + \frac{13}{12} = \frac{25}{12} \]

5. **Final calculation:**
Now substituting back into the expression:
\[ \frac{\frac{13}{5}}{\frac{25}{12}} = \frac{13}{5} \times \frac{12}{25} = \frac{13 \times 12}{5 \times 25} = \frac{156}{125} \]

Thus, the value of \( \frac{\csc \alpha}{1 + \sec \alpha} \) is \( \frac{156}{125} \).