Solve the following equations
`(i) tan^(-1). ( x-1)/(x - 2) = tan ^(-1). ( x+1)/(x + 2) = pi/4 `
(ii) ` tan^(-1) 2 xx tan^(-1) 3x = pi/4`

Answer

To solve the given equations, we will follow a step-by-step approach.

### (i) Solve the equation:
\[ \tan^{-1}\left(\frac{x-1}{x-2}\right) + \tan^{-1}\left(\frac{x+1}{x+2}\right) = \frac{\pi}{4} \]

**Step 1:** Use the formula for the sum of inverse tangents:
\[ \tan^{-1}(a) + \tan^{-1}(b) = \tan^{-1}\left(\frac{a + b}{1 - ab}\right) \]
Let \( a = \frac{x-1}{x-2} \) and \( b = \frac{x+1}{x+2} \).

**Step 2:** Calculate \( a + b \):
\[ a + b = \frac{x-1}{x-2} + \frac{x+1}{x+2} \]
To combine these fractions, find a common denominator:
\[ = \frac{(x-1)(x+2) + (x+1)(x-2)}{(x-2)(x+2)} \]
Expanding the numerator:
\[ = \frac{(x^2 + 2x - x - 2) + (x^2 - 2x + x - 2)}{(x-2)(x+2)} \]
\[ = \frac{2x^2 - 4}{(x-2)(x+2)} = \frac{2(x^2 - 2)}{(x-2)(x+2)} \]

**Step 3:** Calculate \( ab \):
\[ ab = \left(\frac{x-1}{x-2}\right)\left(\frac{x+1}{x+2}\right) = \frac{(x-1)(x+1)}{(x-2)(x+2)} = \frac{x^2 - 1}{(x-2)(x+2)} \]

**Step 4:** Substitute \( a + b \) and \( ab \) into the formula:
\[ \tan^{-1}\left(\frac{2(x^2 - 2)}{(x-2)(x+2) - (x^2 - 1)}\right) = \frac{\pi}{4} \]

**Step 5:** Since \( \tan\left(\frac{\pi}{4}\right) = 1 \), we set the argument equal to 1:
\[ \frac{2(x^2 - 2)}{(x-2)(x+2) - (x^2 - 1)} = 1 \]

**Step 6:** Cross-multiply and simplify:
\[ 2(x^2 - 2) = (x-2)(x+2) - (x^2 - 1) \]
\[ 2x^2 - 4 = x^2 - 4 - x^2 + 1 \]
\[ 2x^2 - 4 = -3 \]
\[ 2x^2 = 1 \implies x^2 = \frac{1}{2} \implies x = \pm \frac{1}{\sqrt{2}} = \pm \frac{\sqrt{2}}{2} \]

### (ii) Solve the equation:
\[ \tan^{-1}(2) + \tan^{-1}(3x) = \frac{\pi}{4} \]

**Step 1:** Use the sum formula for inverse tangents:
\[ \tan^{-1}(2) + \tan^{-1}(3x) = \tan^{-1}\left(\frac{2 + 3x}{1 - 2 \cdot 3x}\right) \]

**Step 2:** Set the argument equal to 1:
\[ \frac{2 + 3x}{1 - 6x} = 1 \]

**Step 3:** Cross-multiply:
\[ 2 + 3x = 1 - 6x \]

**Step 4:** Combine like terms:
\[ 3x + 6x = 1 - 2 \]
\[ 9x = -1 \implies x = -\frac{1}{9} \]

### Final Solutions:
1. For the first equation: \( x = \pm \frac{\sqrt{2}}{2} \)
2. For the second equation: \( x = -\frac{1}{9} \)