The minimum value of x which satisfies the inequality `(sin^(-1)x)^(2)ge(cos^(-1)x)^(2)` is

Options

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

Answer

To solve the inequality \((\sin^{-1} x)^2 \geq (\cos^{-1} x)^2\), we can follow these steps:

### Step 1: Rewrite the Inequality
We start by rewriting the inequality:
\[ (\sin^{-1} x)^2 - (\cos^{-1} x)^2 \geq 0 \]
This can be factored using the difference of squares:
\[ (\sin^{-1} x - \cos^{-1} x)(\sin^{-1} x + \cos^{-1} x) \geq 0 \]

### Step 2: Use the Identity
We know that:
\[ \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \]
This means that the second term \((\sin^{-1} x + \cos^{-1} x)\) is always positive for \(x \in [0, 1]\).

### Step 3: Focus on the First Term
Now, we need to analyze the first term:
\[ \sin^{-1} x - \cos^{-1} x \geq 0 \]
This simplifies to:
\[ \sin^{-1} x \geq \cos^{-1} x \]

### Step 4: Solve the Inequality
Using the identity \(\cos^{-1} x = \frac{\pi}{2} - \sin^{-1} x\), we can rewrite the inequality:
\[ \sin^{-1} x \geq \frac{\pi}{2} - \sin^{-1} x \]
Adding \(\sin^{-1} x\) to both sides gives:
\[ 2\sin^{-1} x \geq \frac{\pi}{2} \]
Dividing by 2:
\[ \sin^{-1} x \geq \frac{\pi}{4} \]

### Step 5: Find the Corresponding x Value
Taking the sine of both sides, we have:
\[ x \geq \sin\left(\frac{\pi}{4}\right) \]
Since \(\sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\), we find:
\[ x \geq \frac{1}{\sqrt{2}} \]

### Conclusion
The minimum value of \(x\) that satisfies the inequality is:
\[ \boxed{\frac{1}{\sqrt{2}}} \]