Answer: (1/2) tan-1 x.
- A (1/2) tan-1 x
- B 2 tan-1 x
- C tan-1 x
- D (1/2) tan-1(1/x)
Correct answer: A. (1/2) tan-1 x
Explanation: Substituting x = tan(theta) with theta in (0,pi/2), (sqrt(1+x<sup>2</sup>)-1)/x = (sec(theta)-1)/tan(theta) = tan(theta/2), so the expression equals theta/2 = (1/2)tan-1 x.
The graph of sin⁻¹x is confined to a narrow domain [-1,1] (since sine itself only takes values in that range) and range [-π/2,π/2] (the principal value branch chosen to make sine one-one and invertible there).
Concept context
Restricting trig functions to make them invertible, the principal value branches of sin-inverse, cos-inverse, tan-inverse, and friends, and the key identities relating them.