If `sin^(-1)(tan(pi/4))-sin^(-1)(sqrt(3/y))-(pi)/6=0` and `x^(2)=y` then `x` is equal to

Options

A. `2`
B. `4`
C. `sqrt(2)`
D. `sqrt(3)`

Explanation

To solve the equation \( \sin^{-1}(\tan(\frac{\pi}{4})) - \sin^{-1}(\sqrt{\frac{3}{y}}) - \frac{\pi}{6} = 0 \) and find the value of \( x \) given that \( x^2 = y \), we can follow these steps:

### Step 1: Simplify \( \tan(\frac{\pi}{4}) \)
We know that:
\[ \tan\left(\frac{\pi}{4}\right) = 1 \]
Thus, we can rewrite the equation as:
\[ \sin^{-1}(1) - \sin^{-1}\left(\sqrt{\frac{3}{y}}\right) - \frac{\pi}{6} = 0 \]

### Step 2: Evaluate \( \sin^{-1}(1) \)
The value of \( \sin^{-1}(1) \) is:
\[ \sin^{-1}(1) = \frac{\pi}{2} \]
Now, substituting this back into the equation gives:
\[ \frac{\pi}{2} - \sin^{-1}\left(\sqrt{\frac{3}{y}}\right) - \frac{\pi}{6} = 0 \]

### Step 3: Rearranging the equation
Rearranging the equation, we have:
\[ \frac{\pi}{2} - \frac{\pi}{6} = \sin^{-1}\left(\sqrt{\frac{3}{y}}\right) \]

### Step 4: Simplifying the left side
To simplify \( \frac{\pi}{2} - \frac{\pi}{6} \), we find a common denominator:
\[ \frac{\pi}{2} = \frac{3\pi}{6} \]
Thus:
\[ \frac{3\pi}{6} - \frac{\pi}{6} = \frac{2\pi}{6} = \frac{\pi}{3} \]
So, we have:
\[ \sin^{-1}\left(\sqrt{\frac{3}{y}}\right) = \frac{\pi}{3} \]

### Step 5: Taking the sine of both sides
Taking the sine of both sides gives:
\[ \sqrt{\frac{3}{y}} = \sin\left(\frac{\pi}{3}\right) \]
We know that:
\[ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]
Thus, we have:
\[ \sqrt{\frac{3}{y}} = \frac{\sqrt{3}}{2} \]

### Step 6: Squaring both sides
Squaring both sides results in:
\[ \frac{3}{y} = \frac{3}{4} \]

### Step 7: Solving for \( y \)
Cross-multiplying gives:
\[ 3 \cdot 4 = 3y \implies 12 = 3y \implies y = 4 \]

### Step 8: Finding \( x \)
Since we are given that \( x^2 = y \):
\[ x^2 = 4 \]
Taking the square root gives:
\[ x = \pm 2 \]

### Final Answer
Thus, the value of \( x \) is:
\[ x = 2 \quad \text{(considering only the positive root as a standard convention)} \]