What is the value of x that satisfies the equation `cos^(-1) x = 2 sin^(-1) x` ?

Options

A. `(1)/(2)`
B. -1
C. 1
D. `-(1)/(2)`

Explanation

To solve the equation \( \cos^{-1} x = 2 \sin^{-1} x \), we can follow these steps:

### Step 1: Rewrite the equation using a trigonometric identity
We know that:
\[ \cos^{-1} x = \frac{\pi}{2} - \sin^{-1} x \]
Substituting this into the equation gives:
\[ \frac{\pi}{2} - \sin^{-1} x = 2 \sin^{-1} x \]

### Step 2: Rearrange the equation
Now, we can rearrange the equation to isolate \( \sin^{-1} x \):
\[ \frac{\pi}{2} = 2 \sin^{-1} x + \sin^{-1} x \]
This simplifies to:
\[ \frac{\pi}{2} = 3 \sin^{-1} x \]

### Step 3: Solve for \( \sin^{-1} x \)
Now, divide both sides by 3:
\[ \sin^{-1} x = \frac{\pi}{6} \]

### Step 4: Find the value of \( x \)
To find \( x \), we take the sine of both sides:
\[ x = \sin\left(\frac{\pi}{6}\right) \]
We know that:
\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]
Thus, we have:
\[ x = \frac{1}{2} \]

### Final Answer
The value of \( x \) that satisfies the equation is:
\[ \boxed{\frac{1}{2}} \]