The smallest value of `3(tan^(2)theta+cot^(2)theta) - 8(tantheta + cottheta)+10, theta in (0,pi/2)` can be

Answer

To find the smallest value of the expression \(3(\tan^2 \theta + \cot^2 \theta) - 8(\tan \theta + \cot \theta) + 10\) for \(\theta \in (0, \frac{\pi}{2})\), we can follow these steps:

### Step 1: Apply the AM-GM Inequality
We start by applying the Arithmetic Mean-Geometric Mean (AM-GM) inequality to the terms \(\tan^2 \theta\) and \(\cot^2 \theta\).

\[ \frac{\tan^2 \theta + \cot^2 \theta}{2} \geq \sqrt{\tan^2 \theta \cdot \cot^2 \theta} \]

Since \(\tan \theta \cdot \cot \theta = 1\), we have:

\[ \sqrt{\tan^2 \theta \cdot \cot^2 \theta} = 1 \implies \tan^2 \theta + \cot^2 \theta \geq 2 \]

### Step 2: Apply AM-GM to \(\tan \theta\) and \(\cot \theta\)
Next, we apply the AM-GM inequality to \(\tan \theta\) and \(\cot \theta\):

\[ \frac{\tan \theta + \cot \theta}{2} \geq \sqrt{\tan \theta \cdot \cot \theta} = 1 \implies \tan \theta + \cot \theta \geq 2 \]

### Step 3: Substitute the Minimum Values
Now we substitute the minimum values obtained from the inequalities into the original expression:

1. From \(\tan^2 \theta + \cot^2 \theta \geq 2\), we have:
\[ 3(\tan^2 \theta + \cot^2 \theta) \geq 3 \cdot 2 = 6 \]

2. From \(\tan \theta + \cot \theta \geq 2\), we have:
\[ -8(\tan \theta + \cot \theta) \leq -8 \cdot 2 = -16 \]

### Step 4: Combine the Results
Now, we can combine these results into the original expression:

\[ 3(\tan^2 \theta + \cot^2 \theta) - 8(\tan \theta + \cot \theta) + 10 \geq 6 - 16 + 10 \]

Calculating this gives:

\[ 6 - 16 + 10 = 0 \]

### Conclusion
Thus, the smallest value of the expression \(3(\tan^2 \theta + \cot^2 \theta) - 8(\tan \theta + \cot \theta) + 10\) is:

\[ \boxed{0} \]