If `a. 3^(tanx) + a. 3^(-tanx) - 2 = 0` has real solutions, `x != pi/2 , 0 le x le pi,` then find the set of all possible values of parameter 'a'.

Options

A. [-1, 1]
B. [-1, 0]
C. (0, 1]
D. `(0, oo)`

Explanation

To solve the equation \( a \cdot 3^{\tan x} + a \cdot 3^{-\tan x} - 2 = 0 \) for real solutions where \( x \neq \frac{\pi}{2} \) and \( 0 \leq x \leq \pi \), we will follow these steps:

### Step 1: Substitute \( y = 3^{\tan x} \)
We start by letting \( y = 3^{\tan x} \). This gives us:
\[ a \cdot y + a \cdot \frac{1}{y} - 2 = 0 \]
This simplifies to:
\[ a y + \frac{a}{y} - 2 = 0 \]

### Step 2: Multiply through by \( y \)
To eliminate the fraction, we multiply the entire equation by \( y \):
\[ a y^2 - 2y + a = 0 \]

### Step 3: Identify the quadratic function
We can express this as a quadratic function:
\[ f(y) = a y^2 - 2y + a \]

### Step 4: Conditions for real and positive roots
For the quadratic equation \( f(y) = 0 \) to have real and positive roots, we need to satisfy three conditions:

1. **Discriminant Condition**: The discriminant must be non-negative.
2. **Value at Zero Condition**: The value of the function at \( y = 0 \) must be positive.
3. **Vertex Condition**: The abscissa of the vertex must be positive.

### Step 5: Apply the Discriminant Condition
The discriminant \( D \) of the quadratic \( a y^2 - 2y + a \) is given by:
\[ D = b^2 - 4ac = (-2)^2 - 4(a)(a) = 4 - 4a^2 \]
For real roots, we need:
\[ 4 - 4a^2 \geq 0 \implies 1 - a^2 \geq 0 \implies -1 \leq a \leq 1 \]

### Step 6: Apply the Value at Zero Condition
We evaluate \( f(0) \):
\[ f(0) = a \cdot 0^2 - 2 \cdot 0 + a = a \]
For \( f(0) > 0 \):
\[ a > 0 \]

### Step 7: Apply the Vertex Condition
The abscissa of the vertex is given by:
\[ x_v = -\frac{b}{2a} = \frac{2}{2a} = \frac{1}{a} \]
For the vertex to be positive:
\[ \frac{1}{a} > 0 \implies a > 0 \]

### Step 8: Combine the Conditions
From the discriminant condition, we have:
\[ -1 \leq a \leq 1 \]
From the value at zero and vertex conditions, we have:
\[ a > 0 \]
Combining these, we find:
\[ 0 < a \leq 1 \]

### Final Answer
The set of all possible values of the parameter \( a \) is:
\[ \boxed{(0, 1]} \]