The trigonometric equation `sin^(-1)x=2sin^(-1)a` has a solution for all real values (b) `|a|<1/a` `|a|lt=1/(sqrt(2))` (d) `1/2<|a|<1/(sqrt(2))`

Explanation

To solve the equation \( \sin^{-1} x = 2 \sin^{-1} a \) and determine the values of \( a \) for which this equation has solutions, we can follow these steps:

### Step 1: Understand the Range of \( \sin^{-1} x \)
The function \( \sin^{-1} x \) (the inverse sine function) has a range of \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \). This means that \( 2 \sin^{-1} a \) must also lie within this range.

### Step 2: Set Up the Inequality
Since \( \sin^{-1} x = 2 \sin^{-1} a \), we can write:
\[
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