Answer: x = pi/4.
- A x = pi/4
- B x = pi/6
- C x = pi/3
- D x = pi/2
Correct answer: A. x = pi/4
Explanation: At x = pi/4: cos(pi/4) = 1/sqrt2, cosec(pi/4) = sqrt2. LHS: 2tan-1(1/sqrt2). RHS: tan-1(2sqrt2). Using the double angle formula 2tan-1(1/sqrt2) = tan-1[2(1/sqrt2)/(1-1/2)] = tan-1[sqrt2/(1/2)] = tan-1(2sqrt2), matching RHS, confirming x=pi/4 works.
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.