Solve for x: sin⁻¹(x) + sin⁻¹(√(1-x²)) = π/2 for x ∈ [0,1].
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Step-by-Step Solution
Step 1: Substitute a trigonometric identity
We are given the equation sin−1(x)+sin−1(1−x2)=2π. For x∈[0,1], we know that 1−x2 is also in [0,1]. A fundamental trigonometric identity states that sin−1(1−x2)=cos−1(x) when x∈[0,1]. This substitution simplifies the equation.
Step 2: Substitute into the original equation
Now, substitute the identity sin−1(1−x2)=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)=2π holds true for all x∈[−1,1]. Since the problem specifies x∈[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)=2π 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].