The most general of `theta` satisfying `tantheta+tan((3pi)/(4)theta)=2` are given by

Answer

To solve the equation \( \tan \theta + \tan \left( \frac{3\pi}{4} \theta \right) = 2 \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \tan \theta + \tan \left( \frac{3\pi}{4} \theta \right) = 2 \]
Using the identity for tangent of a sum, we can express \( \tan \left( \frac{3\pi}{4} \theta \right) \):
\[ \tan \left( \frac{3\pi}{4} \theta \right) = \tan \left( \pi - \frac{\pi}{4} + \theta \right) = -\tan \left( \frac{\pi}{4} - \theta \right) \]

### Step 2: Substitute the identity
Substituting this back into the equation gives:
\[ \tan \theta - \tan \left( \frac{\pi}{4} - \theta \right) = 2 \]

### Step 3: Use the tangent subtraction formula
Using the tangent subtraction formula:
\[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]
We can set \( A = \theta \) and \( B = \frac{\pi}{4} \):
\[ \tan \left( \frac{\pi}{4} - \theta \right) = \frac{1 - \tan \theta}{1 + \tan \theta} \]
Substituting this into our equation:
\[ \tan \theta - \frac{1 - \tan \theta}{1 + \tan \theta} = 2 \]

### Step 4: Clear the fraction
Multiply through by \( 1 + \tan \theta \) to eliminate the fraction:
\[ \tan \theta (1 + \tan \theta) - (1 - \tan \theta) = 2(1 + \tan \theta) \]
This simplifies to:
\[ \tan^2 \theta + \tan \theta - 1 + \tan \theta = 2 + 2\tan \theta \]
Combining like terms gives:
\[ \tan^2 \theta - 1 = 2 \]

### Step 5: Rearrange the equation
Rearranging the equation gives:
\[ \tan^2 \theta - 3 = 0 \]

### Step 6: Solve for \( \tan \theta \)
This can be factored as:
\[ \tan^2 \theta = 3 \]
Taking the square root of both sides, we find:
\[ \tan \theta = \pm \sqrt{3} \]

### Step 7: Find the general solutions
The general solutions for \( \tan \theta = \sqrt{3} \) and \( \tan \theta = -\sqrt{3} \) are:
\[ \theta = n\pi + \frac{\pi}{3} \quad \text{and} \quad \theta = n\pi - \frac{\pi}{3} \quad \text{where } n \in \mathbb{Z} \]

### Final Answer
Thus, the most general solutions for \( \theta \) are:
\[ \theta = n\pi + \frac{\pi}{3} \quad \text{and} \quad \theta = n\pi - \frac{\pi}{3}, \quad n \in \mathbb{Z} \]