If `4\ cos^(-1)x+sin^(-1)x=pi`, then the value of `x` is

Options

A. `3/2`
B. `1/(sqrt(2))`
C. `(sqrt(3))/2`
D. `2/(sqrt(3))`

Explanation

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

### Step 1: Rewrite the equation
We start with the equation:
\[ 4 \cos^{-1} x + \sin^{-1} x = \pi \]
We know that \( \sin^{-1} x \) can be expressed in terms of \( \cos^{-1} x \):
\[ \sin^{-1} x = \frac{\pi}{2} - \cos^{-1} x \]
Substituting this into the equation gives:
\[ 4 \cos^{-1} x + \left( \frac{\pi}{2} - \cos^{-1} x \right) = \pi \]

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

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

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

### Step 5: Find \( x \)
To find \( x \), we take the cosine of both sides:
\[ x = \cos\left(\frac{\pi}{6}\right) \]
We know that:
\[ \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \]
Thus, the value of \( x \) is:
\[ x = \frac{\sqrt{3}}{2} \]

### Final Answer
The value of \( x \) is \( \frac{\sqrt{3}}{2} \).

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