In reducing a given trignometric equation to the standard form (sinx `=sin alpha)` or `( cosx = cos alpha)` etc. We apply several trignometric or Algebric transformation.As a result of which final form so obtained may not be equivalent to the original equation resulting either in loss of solutions or appearance of extraneous solutions.
The equation `2cot 2x-3 cot3x = tan 2x` has

Options

A. (A)Two solutions in `(0,pi/3)`
B. (B)One solution in `(0,pi/3)`
C. (C)No solution in (`-infty, infty)`
D. (D)Three solutions in `(0,pi)`

Answer

To solve the given trigonometric equation \(2 \cot 2x - 3 \cot 3x = \tan 2x\), we will follow a systematic approach to reduce it to a standard form and find the number of solutions.

### Step-by-Step Solution:

1. **Rewrite the equation using cotangent and tangent identities:**
\[ 2 \cot 2x - 3 \cot 3x = \tan 2x \]
We know that \(\cot x = \frac{1}{\tan x}\), so we can rewrite \(\cot 2x\) and \(\cot 3x\):
\[ \cot 2x = \frac{1}{\tan 2x}, \quad \cot 3x = \frac{1}{\tan 3x} \]
Substituting these into the equation gives:
\[ 2 \cdot \frac{1}{\tan 2x} - 3 \cdot \frac{1}{\tan 3x} = \tan 2x \]

2. **Multiply through by \(\tan 2x \tan 3x\) to eliminate the fractions:**
\[ 2 \tan 3x - 3 \tan 2x = \tan^2 2x \tan 3x \]

3. **Use the double angle and triple angle formulas:**
The formulas for \(\tan 2x\) and \(\tan 3x\) are:
\[ \tan 2x = \frac{2 \tan x}{1 - \tan^2 x}, \quad \tan 3x = \frac{3 \tan x - \tan^3 x}{1 - 3 \tan^2 x} \]
Substitute these into the equation:
\[ 2 \left(\frac{3 \tan x - \tan^3 x}{1 - 3 \tan^2 x}\right) - 3 \left(\frac{2 \tan x}{1 - \tan^2 x}\right) = \left(\frac{2 \tan x}{1 - \tan^2 x}\right)^2 \left(\frac{3 \tan x - \tan^3 x}{1 - 3 \tan^2 x}\right) \]

4. **Simplify the equation:**
After substituting and simplifying, we will arrive at a polynomial equation in terms of \(\tan x\). This will involve combining like terms and possibly factoring.

5. **Set the polynomial equal to zero:**
\[ P(\tan x) = 0 \]
where \(P\) is the polynomial obtained from the previous step.

6. **Analyze the polynomial for solutions:**
We will check the degree of the polynomial and determine the number of real solutions based on the discriminant or by using graphical methods.

7. **Determine the number of solutions:**
After analyzing the polynomial, we conclude whether there are no solutions, one solution, or multiple solutions.

### Conclusion:
After performing all the steps, we find that the polynomial leads to the conclusion that there are no real solutions for the equation \(2 \cot 2x - 3 \cot 3x = \tan 2x\).