The least value of `tan^(2)x+cot^(2)x` is :

Options

A. 3
B. 2
C. 0
D. 1

Answer

To find the least value of \( \tan^2 x + \cot^2 x \), we can use the properties of trigonometric functions and some algebraic manipulation.

### Step-by-Step Solution:

1. **Understanding the Expression**:
We need to minimize the expression \( \tan^2 x + \cot^2 x \).

2. **Using the Identity**:
Recall that \( \cot x = \frac{1}{\tan x} \). Therefore, we can rewrite \( \cot^2 x \) as:
\[ \cot^2 x = \frac{1}{\tan^2 x} \]
This allows us to express the original equation as:
\[ y = \tan^2 x + \frac{1}{\tan^2 x} \]

3. **Letting \( t = \tan^2 x \)**:
Substitute \( t \) for \( \tan^2 x \):
\[ y = t + \frac{1}{t} \]

4. **Finding the Minimum Value**:
To find the minimum value of \( y \), we can use calculus or the AM-GM inequality. Using the AM-GM inequality:
\[ y = t + \frac{1}{t} \geq 2\sqrt{t \cdot \frac{1}{t}} = 2 \]
The equality holds when \( t = 1 \), which means \( \tan^2 x = 1 \) or \( x = \frac{\pi}{4} + n\pi \) for any integer \( n \).

5. **Conclusion**:
Therefore, the least value of \( \tan^2 x + \cot^2 x \) is:
\[ \boxed{2} \]