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

Options

A. 2
B. 1
C. `1/3`
D. `1/2`

Answer

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 \(\sin^{-1} x\) and \(\cos^{-1} x\)
We know that:
\[ \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \]
Using this identity, we can rewrite \(\cos^{-1} x\) in terms of \(\sin^{-1} x\):
\[ \cos^{-1} x = \frac{\pi}{2} - \sin^{-1} x \]

### Step 3: Substitute \(\cos^{-1} x\) in the equation
Substituting this into our original equation gives:
\[ 4 \sin^{-1} x + \left(\frac{\pi}{2} - \sin^{-1} x\right) = \pi \]

### Step 4: Simplify the equation
Now, we can simplify the equation:
\[ 4 \sin^{-1} x + \frac{\pi}{2} - \sin^{-1} x = \pi \]
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 5: Isolate \(\sin^{-1} x\)
Now, 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 6: Solve for \(\sin^{-1} x\)
Now, divide both sides by 3:
\[ \sin^{-1} x = \frac{\pi}{6} \]

### Step 7: Find \(x\)
To find \(x\), we take the sine of both sides:
\[ x = \sin\left(\frac{\pi}{6}\right) \]
We know that:
\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]

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