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Regarding the properties of inverse circular functions, which of the following identities is correct (with its stated domain)?

1. sin(sin⁻¹ x) = x for x in [-1, 1]
2. sin⁻¹(sin x) = x for all real x
3. cos(cos⁻¹ x) = x for all real x
4. tan(tan⁻¹ x) = x only for x in [-1, 1]

Correct answer: sin(sin⁻¹ x) = x for x in [-1, 1]

Solution

For inverse trig functions, the identity f(f⁻¹(x)) = x holds whenever x lies in the domain of f⁻¹. Since sin⁻¹ x is defined for x in [-1, 1] and its output is an angle whose sine returns x, sin(sin⁻¹ x) = x for all x in [-1, 1]. The other statements are false: sin⁻¹(sin x) = x only for x in [-pi/2, pi/2]; cos(cos⁻¹ x) = x only for x in [-1, 1]; tan(tan⁻¹ x) = x for all real x.

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