If `"tan"^(-1)(x+1)+cot^(-1)(x-1)="sin"^(-1) (4/5) + cos^(-1) (3/5)`, then x has the value:

Options

A. `4sqrt(3/7)`
B. `4sqrt(7/3)`
C. `14sqrt3`
D. `6sqrt7`

Answer

To solve the equation

\[ \tan^{-1}(x+1) + \cot^{-1}(x-1) = \sin^{-1}\left(\frac{4}{5}\right) + \cos^{-1}\left(\frac{3}{5}\right), \]

we can follow these steps:

### Step 1: Simplify the Right Side
We know that

\[ \sin^{-1}(a) + \cos^{-1}(a) = \frac{\pi}{2}. \]

Thus, we can simplify the right side:

\[ \sin^{-1}\left(\frac{4}{5}\right) + \cos^{-1}\left(\frac{3}{5}\right) = \frac{\pi}{2}. \]

### Step 2: Rewrite the Cotangent Inverse
Using the identity \(\cot^{-1}(x) = \frac{\pi}{2} - \tan^{-1}(x)\), we can rewrite the left side:

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

### Step 3: Simplify the Left Side
This simplifies to:

\[ \tan^{-1}(x+1) - \tan^{-1}(x-1) = 0. \]

### Step 4: Set the Arguments Equal
Since the arctangent function is zero when its argument is zero, we have:

\[ \tan^{-1}(x+1) = \tan^{-1}(x-1). \]

This implies:

\[ x + 1 = x - 1. \]

### Step 5: Solve for x
From the equation \(x + 1 = x - 1\), we can simplify:

\[ 1 = -1, \]

which is not possible. Thus, we need to check our previous steps for any mistakes.

### Step 6: Re-evaluate the Equation
Instead of equating the arguments directly, we can equate the tangent of both sides:

\[ \tan(\tan^{-1}(x+1) - \tan^{-1}(x-1)) = 0. \]

This means:

\[ x + 1 = x - 1 \implies 2 = 0, \]

which again leads to a contradiction.

### Step 7: Check for Values
To find the correct values of \(x\), we can analyze the original equation.

We can also use the tangent subtraction formula:

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

This gives:

\[ \tan^{-1}\left(\frac{2}{1 + (x^2 - 1)}\right) = 0. \]

### Step 8: Solve the Equation
Setting the argument of the tangent equal to zero gives:

\[ \frac{2}{x^2} = 0, \]

which leads us to find \(x^2 = 2\).

### Step 9: Final Value of x
Thus, we find:

\[ x = \sqrt{2} \text{ or } x = -\sqrt{2}. \]

However, since we are dealing with inverse trigonometric functions, we take the positive value:

\[ x = 2. \]

### Conclusion
The value of \(x\) is:

\[ \boxed{2}. \]