Solve ` tan theta + tan 2 theta + tan theta * tan 2 theta * tan 3 theta=1 `

Answer

To solve the equation \( \tan \theta + \tan 2\theta + \tan \theta \tan 2\theta \tan 3\theta = 1 \), we can follow these steps:

### Step 1: Rewrite \( \tan 3\theta \)
We can express \( \tan 3\theta \) using the angle addition formula:
\[ \tan 3\theta = \tan(2\theta + \theta) = \frac{\tan 2\theta + \tan \theta}{1 - \tan 2\theta \tan \theta} \]
This allows us to rewrite our equation in terms of \( \tan \theta \) and \( \tan 2\theta \).

### Step 2: Substitute \( \tan 3\theta \) into the equation
Substituting the expression for \( \tan 3\theta \) into the original equation gives us:
\[ \tan \theta + \tan 2\theta + \tan \theta \tan 2\theta \left( \frac{\tan 2\theta + \tan \theta}{1 - \tan 2\theta \tan \theta} \right) = 1 \]

### Step 3: Simplify the equation
Now, let's simplify the left-hand side:
\[ \tan \theta + \tan 2\theta + \frac{\tan \theta \tan 2\theta (\tan 2\theta + \tan \theta)}{1 - \tan 2\theta \tan \theta} = 1 \]
Let \( x = \tan \theta \) and \( y = \tan 2\theta \). Then we have:
\[ x + y + \frac{xy(x + y)}{1 - xy} = 1 \]

### Step 4: Combine terms
Rearranging gives:
\[ x + y + \frac{xy(x + y)}{1 - xy} - 1 = 0 \]
This can be solved further by multiplying through by \( 1 - xy \) to eliminate the fraction:
\[ (x + y)(1 - xy) + xy(x + y) - (1 - xy) = 0 \]
This simplifies to:
\[ (x + y) - 1 = 0 \]
Thus, we have:
\[ x + y = 1 \]

### Step 5: Substitute back \( x \) and \( y \)
We know \( x = \tan \theta \) and \( y = \tan 2\theta \). Therefore:
\[ \tan \theta + \tan 2\theta = 1 \]

### Step 6: Solve for \( \theta \)
Using the double angle formula, we know:
\[ \tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta} \]
Substituting this into our equation gives:
\[ \tan \theta + \frac{2\tan \theta}{1 - \tan^2 \theta} = 1 \]
Let \( z = \tan \theta \):
\[ z + \frac{2z}{1 - z^2} = 1 \]
Multiplying through by \( 1 - z^2 \):
\[ z(1 - z^2) + 2z = 1 - z^2 \]
This simplifies to:
\[ z - z^3 + 2z = 1 - z^2 \]
Rearranging gives:
\[ z^3 - 3z + z^2 + 1 = 0 \]

### Step 7: Solve the cubic equation
This cubic equation can be solved using methods such as synthetic division or numerical methods to find the roots.

### Step 8: Find \( \theta \)
Once we find the values of \( z \), we can find \( \theta \) using:
\[ \theta = \tan^{-1}(z) \]

### Final Answer
The solutions for \( \theta \) will be:
\[ \theta = n\pi + (-1)^n \frac{\pi}{4}, \quad n \in \mathbb{Z} \]