To solve the equation \( \arctan(x-1) + \arctan(x+1) = \arctan(3x) \), we'll use the formula for the sum of two arctangents:

\[
\arctan(A) + \arctan(B) = \arctan\left(\frac{A + B}{1 - AB}\right), \quad \text{provided } AB \neq 1
\]

### Step 1: Apply the Sum Formula

Given \( A = x-1 \) and \( B = x+1 \):

\[
\arctan(x-1) + \arctan(x+1) = \arctan\left(\frac{(x-1) + (x+1)}{1 - (x-1)(x+1)}\right)
\]

### Step 2: Simplify the Expression

Calculate the numerator:

\[
(x-1) + (x+1) = 2x
\]

Next, calculate the denominator:

\[
1 - (x-1)(x+1) = 1 - (x^2 - 1) = 2 - x^2
\]

Thus:

\[
\arctan(x-1) + \arctan(x+1) = \arctan\left(\frac{2x}{2 - x^2}\right)
\]

### Step 3: Set the Expression Equal to \( \arctan(3x) \)

Now, we equate the two arctangents:

\[
\arctan\left(\frac{2x}{2 - x^2}\right) = \arctan(3x)
\]

Since the arctangent function is injective (one-to-one) within its domain:

\[
\frac{2x}{2 - x^2} = 3x
\]

### Step 4: Solve the Equation

\[
\frac{2x}{2 - x^2} = 3x
\]

Cross-multiply to clear the fraction:

\[
2x = 3x(2 - x^2)
\]

\[
2x = 6x - 3x^3
\]

Rearrange all terms to one side:

\[
3x^3 - 4x = 0
\]

### Step 5: Factor the Equation

\[
x(3x^2 - 4) = 0
\]

This gives us two factors:

1. \( x = 0 \)
2. \( 3x^2 - 4 = 0 \)

### Step 6: Solve Each Factor

1. \( x = 0 \)

2. Solve \( 3x^2 - 4 = 0 \):

\[
3x^2 = 4
\]

\[
x^2 = \frac{4}{3}
\]

\[
x = \pm \frac{2}{\sqrt{3}} = \pm \frac{2\sqrt{3}}{3}
\]

### Final Solutions

The solutions are:

\[
x = 0, \quad x = \frac{2\sqrt{3}}{3}, \quad x = -\frac{2\sqrt{3}}{3}
\]

Would you like more details or have any questions?

### Further Questions

1. What is the geometric interpretation of the solutions in the arctangent equation?
2. How would the solution set change if we modified the constants in the equation?
3. Can you derive the sum of two arctangents formula using trigonometric identities?
4. What is the domain of each solution for \( x \) in the context of this equation?
5. How can we verify these solutions graphically?

### Tip

Always check if the solutions fall within the domain where the arctangent function remains one-to-one to confirm the correctness of your results.

Learn how to solve the trigonometric equation arctan(x-1) + arctan(x+1) = arctan(3x) using the sum of arctangents formula. This detailed step-by-step guide covers key concepts in trigonometry and algebra, suitable for students in Grades 11-12.

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