Answer: x = 1/6.
- A x = 1/6
- B x = 1/3
- C x = 1/2
- D x = 1
Correct answer: A. x = 1/6
Explanation: tan-1(2x)+tan-1(3x) = tan-1[(5x)/(1-6x<sup>2</sup>)] = pi/4 implies 5x/(1-6x<sup>2</sup>)=1, so 5x = 1-6x<sup>2</sup>, giving 6x<sup>2</sup>+5x-1=0, (6x-1)(x+1)=0, so x=1/6 (x=-1 is rejected as it gives a negative argument issue with the formula's validity).
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.