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

Options

A. 0
B. `1/2`
C. `-1/2`
D. 1

Answer

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

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

### Step 3: Simplify the equation
Combine like terms:
\[ 4 \sin^{-1} x - \sin^{-1} x + \frac{\pi}{2} = \pi \]
This simplifies to:
\[ 3 \sin^{-1} x + \frac{\pi}{2} = \pi \]

### Step 4: Isolate \( \sin^{-1} x \)
Subtract \( \frac{\pi}{2} \) from both sides:
\[ 3 \sin^{-1} x = \pi - \frac{\pi}{2} \]
This simplifies to:
\[ 3 \sin^{-1} x = \frac{\pi}{2} \]

### Step 5: Solve for \( \sin^{-1} x \)
Divide both sides by 3:
\[ \sin^{-1} x = \frac{\pi}{6} \]

### Step 6: Find \( x \)
Now, take the sine of both sides:
\[ x = \sin\left(\frac{\pi}{6}\right) \]

### Step 7: Calculate \( \sin\left(\frac{\pi}{6}\right) \)
We know that:
\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]
Thus, we find:
\[ x = \frac{1}{2} \]

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