If `sin^(-1)x-cos^(-1)x=pi/6, " then " x=`

Options

A. `1/2`
B. `-1/2`
C. `sqrt3/2`
D. none

Answer

To solve the equation \( \sin^{-1}x - \cos^{-1}x = \frac{\pi}{6} \), we can follow these steps:

### Step 1: Use the identity for inverse trigonometric functions
We know that:
\[ \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \]
From this identity, we can express \( \cos^{-1}x \) in terms of \( \sin^{-1}x \):
\[ \cos^{-1}x = \frac{\pi}{2} - \sin^{-1}x \]

### Step 2: Substitute into the original equation
Substituting \( \cos^{-1}x \) into the original equation gives:
\[ \sin^{-1}x - \left(\frac{\pi}{2} - \sin^{-1}x\right) = \frac{\pi}{6} \]
This simplifies to:
\[ \sin^{-1}x - \frac{\pi}{2} + \sin^{-1}x = \frac{\pi}{6} \]

### Step 3: Combine like terms
Combining the \( \sin^{-1}x \) terms results in:
\[ 2\sin^{-1}x - \frac{\pi}{2} = \frac{\pi}{6} \]

### Step 4: Isolate \( \sin^{-1}x \)
Now, we can isolate \( \sin^{-1}x \) by adding \( \frac{\pi}{2} \) to both sides:
\[ 2\sin^{-1}x = \frac{\pi}{6} + \frac{\pi}{2} \]

### Step 5: Find a common denominator
To add the fractions, we need a common denominator. The common denominator for 6 and 2 is 6:
\[ \frac{\pi}{2} = \frac{3\pi}{6} \]
Thus,
\[ 2\sin^{-1}x = \frac{\pi}{6} + \frac{3\pi}{6} = \frac{4\pi}{6} \]

### Step 6: Simplify the equation
Now we simplify:
\[ 2\sin^{-1}x = \frac{4\pi}{6} \implies \sin^{-1}x = \frac{4\pi}{12} = \frac{2\pi}{6} = \frac{\pi}{3} \]

### Step 7: Solve for \( x \)
To find \( x \), we take the sine of both sides:
\[ x = \sin\left(\frac{\pi}{3}\right) \]
We know that:
\[ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{\frac{\sqrt{3}}{2}} \]