Solve : `tan^(2)x*tan^(2)3x*tan4x=tan^(2)x-tan^(2)3x+tan4x`

Answer

To solve the equation \( \tan^2 x \tan^2 3x \tan 4x = \tan^2 x - \tan^2 3x + \tan 4x \), we can follow these steps:

### Step 1: Rearranging the Equation
We start with the given equation:
\[ \tan^2 x \tan^2 3x \tan 4x = \tan^2 x - \tan^2 3x + \tan 4x \]
We can rearrange it by moving \(\tan 4x\) to the left side:
\[ \tan^2 x \tan^2 3x \tan 4x - \tan 4x = \tan^2 x - \tan^2 3x \]

### Step 2: Factoring Out \(\tan 4x\)
Now, we can factor out \(\tan 4x\) from the left-hand side:
\[ \tan 4x (\tan^2 x \tan^2 3x - 1) = \tan^2 x - \tan^2 3x \]

### Step 3: Dividing Both Sides
Next, we can divide both sides by \(\tan^2 x \tan^2 3x - 1\) (assuming it is not zero):
\[ \tan 4x = \frac{\tan^2 x - \tan^2 3x}{\tan^2 x \tan^2 3x - 1} \]

### Step 4: Applying the Difference of Squares
We know that \(a^2 - b^2 = (a - b)(a + b)\). We can apply this to the numerator:
\[ \tan 4x = \frac{(\tan 3x - \tan x)(\tan 3x + \tan x)}{\tan^2 x \tan^2 3x - 1} \]

### Step 5: Recognizing the Tangent Addition Formula
We can recognize that the right-hand side resembles the tangent addition formula:
\[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \]
Thus, we can express:
\[ \tan 4x = \tan(3x + x) \]

### Step 6: Setting Up the Equation
This leads us to:
\[ \tan 4x = \tan(3x + x) \]
From this, we can equate the angles:
\[ 4x = 3x + x + n\pi \quad \text{(where \(n\) is any integer)} \]

### Step 7: Solving for \(x\)
This simplifies to:
\[ 4x = 4x + n\pi \]
This implies:
\[ 0 = n\pi \]
Thus, \(n = 0\) gives us:
\[ x = \frac{n\pi}{4} \]

### Step 8: Additional Solutions
Now, we also consider the case from the factor we set aside:
\[ 1 - \tan^2 2x = 0 \Rightarrow \tan^2 2x = 1 \Rightarrow \tan 2x = \pm 1 \]
This leads to:
\[ 2x = \frac{\pi}{4} + n\pi \quad \text{or} \quad 2x = \frac{3\pi}{4} + n\pi \]
Thus:
\[ x = \frac{n\pi}{2} + \frac{\pi}{8} \quad \text{or} \quad x = \frac{n\pi}{2} + \frac{3\pi}{8} \]

### Final Solution
Combining both results, we have:
\[ x = \frac{n\pi}{4} \quad \text{or} \quad x = \frac{n\pi}{2} + \frac{\pi}{8} \]