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: Understand the Inequality
We start with the inequality:
\[ (\sin^{-1} x)^2 \geq (\cos^{-1} x)^2 \]
This can be rewritten as:
\[ (\sin^{-1} x)^2 - (\cos^{-1} x)^2 \geq 0 \]

### Step 2: Factor the Expression
Using the difference of squares, we can factor the left-hand side:
\[ (\sin^{-1} x - \cos^{-1} x)(\sin^{-1} x + \cos^{-1} x) \geq 0 \]

### Step 3: Analyze the Terms
We know that:
\[ \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \]
for \(x \in [0, 1]\). Thus, the second term is always positive. Therefore, we need to focus on the first term:
\[ \sin^{-1} x - \cos^{-1} x \geq 0 \]

### Step 4: Simplify the First Term
This simplifies to:
\[ \sin^{-1} x \geq \cos^{-1} x \]
or equivalently:
\[ \sin^{-1} x \geq \frac{\pi}{2} - \sin^{-1} x \]
Adding \(\sin^{-1} x\) to both sides gives:
\[ 2\sin^{-1} x \geq \frac{\pi}{2} \]
Dividing both sides by 2:
\[ \sin^{-1} x \geq \frac{\pi}{4} \]

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

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