If `2 sin^(-1)x - cos^(-1)x = pi/2`, then x is equal to

Options

A. `1/sqrt2`
B. `-1/sqrt2`
C. `(-sqrt3)/(sqrt2)`
D. `(sqrt3)/2`

Answer

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

### Step 1: Rewrite the equation using the identity
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 \(\cos^{-1} x\) in the equation
Now, substitute \(\cos^{-1} x\) into the original equation:
\[ 2 \sin^{-1} x - \left(\frac{\pi}{2} - \sin^{-1} x\right) = \frac{\pi}{2} \]

### Step 3: Simplify the equation
Distributing the negative sign:
\[ 2 \sin^{-1} x - \frac{\pi}{2} + \sin^{-1} x = \frac{\pi}{2} \]
Combine like terms:
\[ 3 \sin^{-1} x - \frac{\pi}{2} = \frac{\pi}{2} \]

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

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

### Step 6: Find \(x\)
Now, take the sine of both sides:
\[ x = \sin\left(\frac{\pi}{3}\right) \]
Using the known value:
\[ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]

### Final Answer
Thus, the value of \(x\) is:
\[ \boxed{\frac{\sqrt{3}}{2}} \]

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