If cos ` A = (9)/(41) `, find the value of
` (1)/(sin ^(2)A ) - cot ^(2) A `

Answer

To solve the problem, we need to find the value of \( \frac{1}{\sin^2 A} - \cot^2 A \) given that \( \cos A = \frac{9}{41} \).

### Step-by-step Solution:

1. **Use the Pythagorean Identity**:
We know that \( \sin^2 A + \cos^2 A = 1 \). Since we have \( \cos A = \frac{9}{41} \), we can find \( \sin^2 A \).
\[ \sin^2 A = 1 - \cos^2 A \]
Calculate \( \cos^2 A \):
\[ \cos^2 A = \left(\frac{9}{41}\right)^2 = \frac{81}{1681} \]
Therefore,
\[ \sin^2 A = 1 - \frac{81}{1681} = \frac{1681 - 81}{1681} = \frac{1600}{1681} \]

2. **Find \( \cot^2 A \)**:
Recall that \( \cot A = \frac{\cos A}{\sin A} \). Thus,
\[ \cot^2 A = \frac{\cos^2 A}{\sin^2 A} \]
Substitute the values:
\[ \cot^2 A = \frac{\frac{81}{1681}}{\frac{1600}{1681}} = \frac{81}{1600} \]

3. **Substitute into the expression**:
Now we substitute \( \sin^2 A \) and \( \cot^2 A \) into the expression \( \frac{1}{\sin^2 A} - \cot^2 A \):
\[ \frac{1}{\sin^2 A} = \frac{1681}{1600} \]
Therefore,
\[ \frac{1}{\sin^2 A} - \cot^2 A = \frac{1681}{1600} - \frac{81}{1600} \]

4. **Combine the fractions**:
Since both fractions have the same denominator, we can combine them:
\[ \frac{1681 - 81}{1600} = \frac{1600}{1600} = 1 \]

Thus, the final answer is:
\[ \frac{1}{\sin^2 A} - \cot^2 A = 1 \]