Let x(0,1).

Question

Let $x\in(0,1)$. The set of all $x$ such that $\sin^{-1}x>\cos^{-1}x$, is the interval:

Accepted Answer

$$\sin^{-1}x>\cos^{-1}x=\frac{\pi}{2}-\sin^{-1}x\ \Rightarrow\ 2\sin^{-1}x>\frac{\pi}{2}\ \Rightarrow\ \sin^{-1}x>\frac{\pi}{4}\ \Rightarrow\ x>\frac{1}{\sqrt2}.$$

Since $\sin^{-1}x$'s maximum on $(0,1)$ approaches but does not reach $\dfrac{\pi}{2}$ (at $x\to1$), the set is $\left(\dfrac{1}{\sqrt2},1\right)$.

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