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 follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \tan \theta + \cot \theta \]
Using the definitions of tangent and cotangent, we can rewrite this as:
\[ \tan \theta + \cot \theta = \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} \]

### Step 2: Find a common denominator
To combine the two fractions, we find a common denominator:
\[ \tan \theta + \cot \theta = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} \]

### Step 3: Use the Pythagorean identity
Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can simplify the expression:
\[ \tan \theta + \cot \theta = \frac{1}{\sin \theta \cos \theta} \]

### Step 4: Express in terms of sine double angle
We know that \( \sin 2\theta = 2 \sin \theta \cos \theta \), so we can rewrite our expression as:
\[ \tan \theta + \cot \theta = \frac{2}{\sin 2\theta} \]

### Step 5: Analyze the function
To find the minimum value of \( \tan \theta + \cot \theta \), we need to maximize \( \sin 2\theta \). The function \( \sin 2\theta \) reaches its maximum value of 1 when \( 2\theta = \frac{\pi}{2} \) or \( \theta = \frac{\pi}{4} \).

### Step 6: Calculate the minimum value
Substituting \( \theta = \frac{\pi}{4} \) into our expression:
\[ \tan \left(\frac{\pi}{4}\right) + \cot \left(\frac{\pi}{4}\right) = 1 + 1 = 2 \]

### Conclusion
Thus, the least value of \( \tan \theta + \cot \theta \) for \( 0 < \theta < \frac{\pi}{2} \) is:
\[ \boxed{2} \]