The possible values of x, which satisfy the trigonometric equation `tan^(-1)((x-1)/(x-2))+tan^(-1)((x+1)/(x+2))= pi/4` are

Options

A. `pm 1/sqrt(2)`
B. `pm sqrt(2)`
C. `pm 1/2`
D. `pm2`

Answer

To solve the equation

\[ \tan^{-1}\left(\frac{x-1}{x-2}\right) + \tan^{-1}\left(\frac{x+1}{x+2}\right) = \frac{\pi}{4}, \]

we can use the formula for the sum of inverse tangents:

\[ \tan^{-1}(a) + \tan^{-1}(b) = \tan^{-1}\left(\frac{a + b}{1 - ab}\right) \]

provided that \(ab < 1\).

### Step 1: Identify \(a\) and \(b\)

Let

\[ a = \frac{x-1}{x-2} \quad \text{and} \quad b = \frac{x+1}{x+2}. \]

### Step 2: Calculate \(a + b\)

We need to find \(a + b\):

\[ a + b = \frac{x-1}{x-2} + \frac{x+1}{x+2}. \]

To combine these fractions, we find a common denominator:

\[ a + b = \frac{(x-1)(x+2) + (x+1)(x-2)}{(x-2)(x+2)}. \]

### Step 3: Simplify the numerator

Expanding the numerator:

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

Adding these two results:

\[ a + b = \frac{(x^2 + x - 2) + (x^2 - x - 2)}{(x-2)(x+2)} = \frac{2x^2 - 4}{(x-2)(x+2)}. \]

### Step 4: Calculate \(ab\)

Now we calculate \(ab\):

\[ ab = \left(\frac{x-1}{x-2}\right) \left(\frac{x+1}{x+2}\right) = \frac{(x-1)(x+1)}{(x-2)(x+2)} = \frac{x^2 - 1}{x^2 - 4}. \]

### Step 5: Apply the formula for the sum of inverse tangents

Using the sum of inverse tangents formula:

\[ \tan^{-1}\left(\frac{2x^2 - 4}{(x-2)(x+2) - (x^2 - 1)}\right) = \frac{\pi}{4}. \]

Since \(\tan\left(\frac{\pi}{4}\right) = 1\), we have:

\[ \frac{2x^2 - 4}{(x-2)(x+2) - (x^2 - 1)} = 1. \]

### Step 6: Set up the equation

Cross-multiplying gives:

\[ 2x^2 - 4 = (x-2)(x+2) - (x^2 - 1). \]

### Step 7: Simplify the right-hand side

Calculating the right-hand side:

\[ (x-2)(x+2) = x^2 - 4, \]
so we have:

\[ x^2 - 4 - (x^2 - 1) = -4 + 1 = -3. \]

### Step 8: Set up the final equation

Thus, we have:

\[ 2x^2 - 4 = -3. \]

Adding 4 to both sides gives:

\[ 2x^2 = 1. \]

### Step 9: Solve for \(x\)

Dividing by 2:

\[ x^2 = \frac{1}{2}. \]

Taking the square root:

\[ x = \pm \frac{1}{\sqrt{2}}. \]

### Conclusion

The possible values of \(x\) that satisfy the equation are:

\[ x = \frac{1}{\sqrt{2}} \quad \text{and} \quad x = -\frac{1}{\sqrt{2}}. \]