If `cot A +(1)/( cot A) =2` , then `cot^2 A + (1)/( cot^2 A)` is equal to

Options

A. 4
B. `sqrt(2)`
C. `1`
D. `2`

Answer

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

1. **Given:**
\[ \cot A + \frac{1}{\cot A} = 2 \]

2. **Square both sides:**
\[ \left(\cot A + \frac{1}{\cot A}\right)^2 = 2^2 \]
This expands to:
\[ \cot^2 A + 2 \cdot \cot A \cdot \frac{1}{\cot A} + \frac{1}{\cot^2 A} = 4 \]

3. **Simplify the equation:**
Since \(\cot A \cdot \frac{1}{\cot A} = 1\), we can simplify the equation:
\[ \cot^2 A + 2 + \frac{1}{\cot^2 A} = 4 \]

4. **Rearranging the equation:**
Subtract 2 from both sides:
\[ \cot^2 A + \frac{1}{\cot^2 A} = 4 - 2 \]
This simplifies to:
\[ \cot^2 A + \frac{1}{\cot^2 A} = 2 \]

5. **Conclusion:**
Therefore, the value of \(\cot^2 A + \frac{1}{\cot^2 A}\) is:
\[ \boxed{2} \]