The solution of the equation `sin^-1((tan)pi/4)-sin^-1(sqrt(3/x))-pi/6=0` is

Options

A. `x=2`
B. `x=-4`
C. `x=4`
D. `x=3`

Explanation

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

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

### Step 3: Combine the terms
Now we can combine \( \frac{\pi}{2} \) and \( -\frac{\pi}{6} \):
\[ \frac{\pi}{2} - \frac{\pi}{6} = \frac{3\pi}{6} - \frac{\pi}{6} = \frac{2\pi}{6} = \frac{\pi}{3} \]
So, we have:
\[ \frac{\pi}{3} - \sin^{-1}\left(\sqrt{\frac{3}{x}}\right) = 0 \]

### Step 4: Isolate \( \sin^{-1}\left(\sqrt{\frac{3}{x}}\right) \)
Rearranging gives:
\[ \sin^{-1}\left(\sqrt{\frac{3}{x}}\right) = \frac{\pi}{3} \]

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

### Step 6: Square both sides
Squaring both sides results in:
\[ \frac{3}{x} = \left(\frac{\sqrt{3}}{2}\right)^2 \]
Calculating the right side gives:
\[ \frac{3}{x} = \frac{3}{4} \]

### Step 7: Cross-multiply to solve for \( x \)
Cross-multiplying gives:
\[ 3 \cdot 4 = 3x \implies 12 = 3x \]
Dividing both sides by 3 results in:
\[ x = 4 \]

### Conclusion
The solution to the equation is:
\[ \boxed{4} \]