`tan^-1(x+1)+cot^-1(1/(x-1))=tan^-1(8/31)` then sum of all the values x satisfying

Answer

To solve the equation \( \tan^{-1}(x+1) + \cot^{-1}\left(\frac{1}{x-1}\right) = \tan^{-1}\left(\frac{8}{31}\right) \), we can follow these steps:

### Step 1: Rewrite the cotangent inverse
Recall that \( \cot^{-1}(y) = \frac{\pi}{2} - \tan^{-1}(y) \). Therefore, we can rewrite the equation as:
\[ \tan^{-1}(x+1) + \left(\frac{\pi}{2} - \tan^{-1}(x-1)\right) = \tan^{-1}\left(\frac{8}{31}\right) \]
This simplifies to:
\[ \tan^{-1}(x+1) - \tan^{-1}(x-1) + \frac{\pi}{2} = \tan^{-1}\left(\frac{8}{31}\right) \]

### Step 2: Isolate the arctangent terms
Rearranging gives us:
\[ \tan^{-1}(x+1) - \tan^{-1}(x-1) = \tan^{-1}\left(\frac{8}{31}\right) - \frac{\pi}{2} \]
Using the identity \( \tan^{-1}(a) - \tan^{-1}(b) = \tan^{-1}\left(\frac{a-b}{1+ab}\right) \), we can express the left side as:
\[ \tan^{-1}\left(\frac{(x+1)-(x-1)}{1+(x+1)(x-1)}\right) = \tan^{-1}\left(\frac{2}{1 + (x^2 - 1)}\right) \]
This simplifies to:
\[ \tan^{-1}\left(\frac{2}{x^2}\right) \]

### Step 3: Set the arctangents equal
Now we have:
\[ \tan^{-1}\left(\frac{2}{x^2}\right) = \tan^{-1}\left(\frac{8}{31}\right) \]
Since the tangent function is one-to-one, we can equate the arguments:
\[ \frac{2}{x^2} = \frac{8}{31} \]

### Step 4: Cross-multiply and solve for \( x^2 \)
Cross-multiplying gives:
\[ 2 \cdot 31 = 8 \cdot x^2 \]
This simplifies to:
\[ 62 = 8x^2 \quad \Rightarrow \quad x^2 = \frac{62}{8} = \frac{31}{4} \]

### Step 5: Solve for \( x \)
Taking the square root gives:
\[ x = \pm \sqrt{\frac{31}{4}} = \pm \frac{\sqrt{31}}{2} \]

### Step 6: Verify the solutions
We need to check if both values satisfy the original equation.

1. For \( x = \frac{\sqrt{31}}{2} \):
- \( x + 1 > 0 \) and \( x - 1 > 0 \) (valid).

2. For \( x = -\frac{\sqrt{31}}{2} \):
- \( x + 1 < 0 \) (invalid as it leads to undefined behavior in the arctangent).

Thus, the only valid solution is \( x = \frac{\sqrt{31}}{2} \).

### Step 7: Calculate the sum of all valid \( x \)
Since the only valid solution is \( x = \frac{\sqrt{31}}{2} \), the sum of all values of \( x \) satisfying the equation is:
\[ \text{Sum} = \frac{\sqrt{31}}{2} \]