Solve for x: sin⁻¹(x) + sin⁻¹(√(1-x²)) = π/2 for x ∈ [0,1].

Step 1: Substitute a trigonometric identity

We are given the equation $\sin^{-1}(x) + \sin^{-1}(\sqrt{1-x^2}) = \frac{\pi}{2}$. For $x \in [0{,}1]$, we know that $\sqrt{1-x^2}$ is also in $[0{,}1]$. A fundamental trigonometric identity states that $\sin^{-1}(\sqrt{1-x^2}) = \cos^{-1}(x)$ when $x \in [0{,}1]$. This substitution simplifies the equation.

Step 2: Substitute into the original equation

Now, substitute the identity $\sin^{-1}(\sqrt{1-x^2}) = \cos^{-1}(x)$ back into the original equation. This gives us a well-known inverse trigonometric identity.

Step 3: Apply the inverse trigonometric identity

The identity $\sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2}$ holds true for all $x \in [-1, 1]$. Since the problem specifies $x \in [0{,}1]$, this identity is valid for the given domain.

Step 4: Determine the solution set

Since the identity $\sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2}$ is true for all $x$ in the domain $[0{,}1]$, any value of $x$ within this interval will satisfy the original equation. Therefore, the solution set is the entire interval $[0{,}1]$.

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