If `tan (sec^(-1)x) =sin (cos^(-1) (1/sqrt5))`, then x is equal to :

Options

A. `+- 3/(sqrt5)`
B. `+- (sqrt5)/3`
C. ` +- (sqrt3)/5`
D. None of these

Answer

To solve the equation \( \tan(\sec^{-1} x) = \sin(\cos^{-1} (1/\sqrt{5})) \), we will follow these steps:

### Step 1: Simplify the Left Side
We start with the left side of the equation:
\[ \tan(\sec^{-1} x) \]
Using the identity \( \tan(\sec^{-1} x) = \sqrt{x^2 - 1} / x \), we can rewrite this as:
\[ \tan(\sec^{-1} x) = \frac{\sqrt{x^2 - 1}}{x} \]

### Step 2: Simplify the Right Side
Now, we simplify the right side:
\[ \sin(\cos^{-1}(1/\sqrt{5})) \]
Using the identity \( \sin(\cos^{-1} \theta) = \sqrt{1 - \theta^2} \), we have:
\[ \sin(\cos^{-1}(1/\sqrt{5})) = \sqrt{1 - (1/\sqrt{5})^2} = \sqrt{1 - \frac{1}{5}} = \sqrt{\frac{4}{5}} = \frac{2}{\sqrt{5}} \]

### Step 3: Set the Two Sides Equal
Now we set the two sides equal:
\[ \frac{\sqrt{x^2 - 1}}{x} = \frac{2}{\sqrt{5}} \]

### Step 4: Cross Multiply
Cross multiplying gives us:
\[ \sqrt{x^2 - 1} \cdot \sqrt{5} = 2x \]

### Step 5: Square Both Sides
Squaring both sides to eliminate the square root:
\[ 5(x^2 - 1) = 4x^2 \]

### Step 6: Rearrange the Equation
Rearranging the equation gives:
\[ 5x^2 - 5 = 4x^2 \]
\[ 5x^2 - 4x^2 - 5 = 0 \]
\[ x^2 - 5 = 0 \]

### Step 7: Solve for \( x^2 \)
This simplifies to:
\[ x^2 = 5 \]

### Step 8: Take the Square Root
Taking the square root of both sides gives:
\[ x = \pm \sqrt{5} \]

### Conclusion
Since \( x \) represents the secant function's input, we only consider the positive value:
\[ x = \sqrt{5} \]