The expression `(1 + tan x + tan^2 x)(1-cot x + cot^2 x)` has the positive values for x, given by

Options

A. `0 le x le (pi)/(2)`
B. `0 le x le pi`
C. `"for all" x in R-{0, pi//2}`
D. `x ge 0`

Answer

To solve the expression \((1 + \tan x + \tan^2 x)(1 - \cot x + \cot^2 x)\) and find the positive values of \(x\) for which this expression is positive, we can follow these steps:

### Step 1: Rewrite the Expression
The expression can be rewritten as:
\[ f(x) = (1 + \tan x + \tan^2 x)(1 - \cot x + \cot^2 x) \]

### Step 2: Expand the Expression
We will expand the expression by multiplying the two parts:
\[ f(x) = (1 + \tan x + \tan^2 x)(1 - \cot x + \cot^2 x) \]
This results in:
\[ = 1 - \cot x + \cot^2 x + \tan x - \tan x \cot x + \tan^2 x - \tan^2 x \cot x + \tan^2 x \cot^2 x \]

### Step 3: Simplify Using Trigonometric Identities
Recall that \(\tan x = \frac{1}{\cot x}\). Using this identity, we can simplify \(\tan^2 x\) and \(\tan x \cot x\):
- \(\tan^2 x = \frac{1}{\cot^2 x}\)
- \(\tan x \cot x = 1\)

Substituting these into the expression gives:
\[ f(x) = 1 - \cot x + \cot^2 x + \tan x - 1 + \tan^2 x - \tan^2 x \cot x + \tan^2 x \cot^2 x \]
This simplifies to:
\[ f(x) = \cot^2 x + \tan^2 x + 1 - \tan^2 x \cot x \]

### Step 4: Analyze the Expression
Now we analyze the expression:
\[ f(x) = 1 + \cot^2 x + \tan^2 x \]
Both \(\cot^2 x\) and \(\tan^2 x\) are always non-negative for all \(x\) where they are defined.

### Step 5: Apply the AM-GM Inequality
Using the AM-GM inequality:
\[ \frac{\cot^2 x + \tan^2 x}{2} \geq 1 \]
This implies:
\[ \cot^2 x + \tan^2 x \geq 2 \]
Thus:
\[ f(x) = 1 + \cot^2 x + \tan^2 x \geq 1 + 2 = 3 \]

### Step 6: Determine Conditions for Positivity
The expression \(f(x) \geq 3\) indicates that \(f(x)\) is always positive except where \(\tan x\) or \(\cot x\) are undefined. This occurs at:
- \(x = 0\) (where \(\cot x\) is undefined)
- \(x = \frac{\pi}{2}\) (where \(\tan x\) is undefined)

### Conclusion
Thus, the expression \((1 + \tan x + \tan^2 x)(1 - \cot x + \cot^2 x)\) has positive values for \(x\) in the intervals:
\[ x \in (0, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \pi) \cup (\pi, \frac{3\pi}{2}) \cup (\frac{3\pi}{2}, 2\pi) \ldots \]
and so on, avoiding points where \(\tan x\) or \(\cot x\) are undefined.