If `4sin^(-1)x+cos^(-1)x=pi`, then `x` is equal to

Options

A. `1/2`
B. `sqrt(3)/(2)`
C. `-1/2`
D. none of these

Explanation

To solve the equation \( 4\sin^{-1}x + \cos^{-1}x = \pi \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 4\sin^{-1}x + \cos^{-1}x = \pi \]

### Step 2: Use the identity for inverse trigonometric functions
We know that:
\[ \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \]
So, we can substitute \(\cos^{-1}x\) in our equation:
\[ 4\sin^{-1}x + \left(\frac{\pi}{2} - \sin^{-1}x\right) = \pi \]

### Step 3: Simplify the equation
Now, we simplify the equation:
\[ 4\sin^{-1}x - \sin^{-1}x + \frac{\pi}{2} = \pi \]
This simplifies to:
\[ 3\sin^{-1}x + \frac{\pi}{2} = \pi \]

### Step 4: Isolate the term with \(\sin^{-1}x\)
Next, we isolate \(3\sin^{-1}x\):
\[ 3\sin^{-1}x = \pi - \frac{\pi}{2} \]
This simplifies to:
\[ 3\sin^{-1}x = \frac{\pi}{2} \]

### Step 5: Divide by 3
Now, we divide both sides by 3:
\[ \sin^{-1}x = \frac{\pi}{6} \]

### Step 6: Solve for \(x\)
To find \(x\), we take the sine of both sides:
\[ x = \sin\left(\frac{\pi}{6}\right) \]
Since \(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\), we have:
\[ x = \frac{1}{2} \]

### Final Answer
Thus, the value of \(x\) is:
\[ \boxed{\frac{1}{2}} \]