If ` 0 lt theta lt pi//2` then .find the least value of `tan theta + cot theta `

Answer

To find the least value of \( \tan \theta + \cot \theta \) for \( 0 < \theta < \frac{\pi}{2} \), we can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-Step Solution:

1. **Apply AM-GM Inequality**:
According to the AM-GM inequality, for any two non-negative numbers \( a \) and \( b \):
\[ \frac{a + b}{2} \geq \sqrt{ab} \]
Here, let \( a = \tan \theta \) and \( b = \cot \theta \).

2. **Set Up the Inequality**:
Substitute \( a \) and \( b \) into the AM-GM inequality:
\[ \frac{\tan \theta + \cot \theta}{2} \geq \sqrt{\tan \theta \cdot \cot \theta} \]

3. **Simplify the Right Side**:
We know that \( \tan \theta \cdot \cot \theta = 1 \) (since \( \cot \theta = \frac{1}{\tan \theta} \)):
\[ \sqrt{\tan \theta \cdot \cot \theta} = \sqrt{1} = 1 \]

4. **Rewrite the Inequality**:
Thus, we have:
\[ \frac{\tan \theta + \cot \theta}{2} \geq 1 \]

5. **Multiply Both Sides by 2**:
Multiplying both sides by 2 gives:
\[ \tan \theta + \cot \theta \geq 2 \]

6. **Conclusion**:
The least value of \( \tan \theta + \cot \theta \) is 2. This minimum occurs when \( \tan \theta = \cot \theta \), which happens when \( \theta = \frac{\pi}{4} \).

### Final Answer:
The least value of \( \tan \theta + \cot \theta \) is \( 2 \).

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