If `sin^(-1)(tan pi//4)-sin^(-1)(sqrt(3//x))-pi//6=0`, then x is a root of the equation

Options

A. `x^(2)-x-6=0`
B. `x^(2)+x-6=0`
C. `x^(2)-x-12=0`
D. `x^(2)+x-12=0`

Answer

To solve the equation \( \sin^{-1}(\tan(\frac{\pi}{4})) - \sin^{-1}\left(\frac{\sqrt{3}}{x}\right) - \frac{\pi}{6} = 0 \), we will 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:
\[ \sin^{-1}(1) - \sin^{-1}\left(\frac{\sqrt{3}}{x}\right) - \frac{\pi}{6} = 0 \]

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

### Step 3: Rearrange the equation
Rearranging gives:
\[ \frac{\pi}{2} - \frac{\pi}{6} = \sin^{-1}\left(\frac{\sqrt{3}}{x}\right) \]

### Step 4: Find a common denominator
To simplify \( \frac{\pi}{2} - \frac{\pi}{6} \), we find a common denominator. The least common multiple of 2 and 6 is 6:
\[ \frac{\pi}{2} = \frac{3\pi}{6} \]
Thus:
\[ \frac{3\pi}{6} - \frac{\pi}{6} = \frac{2\pi}{6} = \frac{\pi}{3} \]
So now we have:
\[ \sin^{-1}\left(\frac{\sqrt{3}}{x}\right) = \frac{\pi}{3} \]

### Step 5: Take the sine of both sides
Taking the sine of both sides gives:
\[ \frac{\sqrt{3}}{x} = \sin\left(\frac{\pi}{3}\right) \]

### Step 6: Evaluate \( \sin\left(\frac{\pi}{3}\right) \)
We know:
\[ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]
Thus:
\[ \frac{\sqrt{3}}{x} = \frac{\sqrt{3}}{2} \]

### Step 7: Solve for \( x \)
Cross-multiplying gives:
\[ \sqrt{3} \cdot 2 = \sqrt{3} \cdot x \]
Dividing both sides by \( \sqrt{3} \) (assuming \( \sqrt{3} \neq 0 \)):
\[ 2 = x \]

### Step 8: Conclusion
Thus, we find:
\[ x = 2 \]

### Step 9: Check for the root of the equation
We need to check which equation \( x = 2 \) is a root of. We will substitute \( x = 2 \) into the given options (not provided in the question, but we can assume they are quadratic or polynomial equations).