If tan A + cot A = 5, find the value of `tan^2 A + cot^2 A`.

Answer

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

1. **Given Equation**:
\[ \tan A + \cot A = 5 \]

2. **Square Both Sides**:
We square both sides of the equation:
\[ (\tan A + \cot A)^2 = 5^2 \]
This expands to:
\[ \tan^2 A + 2 \tan A \cot A + \cot^2 A = 25 \]

3. **Use the Identity**:
We know that \(\tan A \cot A = 1\). Therefore, \(2 \tan A \cot A = 2\):
\[ \tan^2 A + 2 + \cot^2 A = 25 \]

4. **Rearranging the Equation**:
Now, we can rearrange the equation to isolate \(\tan^2 A + \cot^2 A\):
\[ \tan^2 A + \cot^2 A = 25 - 2 \]
\[ \tan^2 A + \cot^2 A = 23 \]

5. **Final Answer**:
Therefore, the value of \(\tan^2 A + \cot^2 A\) is:
\[ \boxed{23} \]