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

Options

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

Answer

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

### Step 1: Rewrite the equation
Start with the given equation:
\[ \cos^{-1}(x) = 2 \sin^{-1}(x) \]

### Step 2: Use the identity for cosine
Recall the identity that relates cosine and sine:
\[ \cos^{-1}(x) = \frac{\pi}{2} - \sin^{-1}(x) \]
Substituting this into the equation gives:
\[ \frac{\pi}{2} - \sin^{-1}(x) = 2 \sin^{-1}(x) \]

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

### Step 4: Solve for \( \sin^{-1}(x) \)
Now, divide both sides by 3:
\[ \sin^{-1}(x) = \frac{\pi}{6} \]

### Step 5: Find \( x \)
To find \( x \), 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:
\[ x = \frac{1}{2} \]

### Conclusion
The value of \( x \) that satisfies the equation \( \cos^{-1}(x) = 2 \sin^{-1}(x) \) is:
\[ \boxed{\frac{1}{2}} \]