If: `tan^(-1) ((sqrt(1 + x^2)-1)/(x)) = 4` then : x =

Options

A. `tan 2`
B. `tan 4`
C. `tan (1/4)`
D. `tan 8`

Answer

To solve the equation \( \tan^{-1} \left( \frac{\sqrt{1 + x^2} - 1}{x} \right) = 4 \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \tan^{-1} \left( \frac{\sqrt{1 + x^2} - 1}{x} \right) = 4 \]
Taking the tangent of both sides, we have:
\[ \frac{\sqrt{1 + x^2} - 1}{x} = \tan(4) \]

### Step 2: Cross-multiply
Now, we can cross-multiply to eliminate the fraction:
\[ \sqrt{1 + x^2} - 1 = x \tan(4) \]

### Step 3: Isolate the square root
Next, we isolate the square root:
\[ \sqrt{1 + x^2} = x \tan(4) + 1 \]

### Step 4: Square both sides
Now, we square both sides to eliminate the square root:
\[ 1 + x^2 = (x \tan(4) + 1)^2 \]

### Step 5: Expand the right side
Expanding the right side gives:
\[ 1 + x^2 = x^2 \tan^2(4) + 2x \tan(4) + 1 \]

### Step 6: Simplify the equation
Subtracting 1 from both sides:
\[ x^2 = x^2 \tan^2(4) + 2x \tan(4) \]
Rearranging gives:
\[ x^2 - x^2 \tan^2(4) - 2x \tan(4) = 0 \]

### Step 7: Factor the equation
Factoring out \( x \):
\[ x (1 - \tan^2(4)) - 2 \tan(4) = 0 \]

### Step 8: Solve for x
Setting each factor to zero gives us:
1. \( x = 0 \) (not a valid solution since it would make the original equation undefined)
2. \( 1 - \tan^2(4) = 2 \tan(4) \)

From \( 1 - \tan^2(4) = 2 \tan(4) \), we can rearrange it to:
\[ 1 = \tan^2(4) + 2 \tan(4) \]
This is a quadratic equation in terms of \( \tan(4) \):
\[ \tan^2(4) + 2 \tan(4) - 1 = 0 \]

### Step 9: Use the quadratic formula
Using the quadratic formula \( \tan(4) = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ \tan(4) = \frac{-2 \pm \sqrt{2^2 - 4(1)(-1)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 4}}{2} = \frac{-2 \pm \sqrt{8}}{2} = \frac{-2 \pm 2\sqrt{2}}{2} = -1 \pm \sqrt{2} \]

### Step 10: Find x
Since \( x = \tan(A) \) where \( A = 4 \), we substitute back to find \( x \):
\[ x = -1 + \sqrt{2} \quad \text{(since } \tan(4) \text{ must be positive)} \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{-1 + \sqrt{2}} \]