In `tanA=5/12` then the value of `(cosA-sinA)` `cosec A` is…….

Options

A. `2/3`
B. `7/5`
C. `3/2`
D. `5/8`

Answer

To solve the problem where \( \tan A = \frac{5}{12} \) and we need to find the value of \( ( \cos A - \sin A ) \csc A \), we can follow these steps:

### Step 1: Understand the relationship of tangent
Given that \( \tan A = \frac{5}{12} \), we can interpret this in terms of a right triangle where:
- The opposite side (perpendicular) to angle \( A \) is 5.
- The adjacent side (base) to angle \( A \) is 12.

### Step 2: Use the Pythagorean theorem to find the hypotenuse
Using the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Opposite}^2 + \text{Adjacent}^2 \]
\[ \text{Hypotenuse}^2 = 5^2 + 12^2 = 25 + 144 = 169 \]
\[ \text{Hypotenuse} = \sqrt{169} = 13 \]

### Step 3: Calculate \( \cos A \) and \( \sin A \)
Now we can find \( \cos A \) and \( \sin A \):
- \( \cos A = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{12}{13} \)
- \( \sin A = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{5}{13} \)

### Step 4: Calculate \( \csc A \)
The cosecant function is the reciprocal of the sine function:
\[ \csc A = \frac{1}{\sin A} = \frac{1}{\frac{5}{13}} = \frac{13}{5} \]

### Step 5: Substitute values into the expression
Now we substitute \( \cos A \), \( \sin A \), and \( \csc A \) into the expression \( ( \cos A - \sin A ) \csc A \):
\[ ( \cos A - \sin A ) \csc A = \left( \frac{12}{13} - \frac{5}{13} \right) \cdot \frac{13}{5} \]

### Step 6: Simplify the expression
Calculate \( \cos A - \sin A \):
\[ \frac{12}{13} - \frac{5}{13} = \frac{12 - 5}{13} = \frac{7}{13} \]
Now multiply by \( \csc A \):
\[ \left( \frac{7}{13} \right) \cdot \left( \frac{13}{5} \right) = \frac{7 \cdot 13}{13 \cdot 5} = \frac{7}{5} \]

### Final Answer
Thus, the value of \( ( \cos A - \sin A ) \csc A \) is:
\[ \frac{7}{5} \]