Question

Evaluate the following:

`tan^-1 sqrt(3) - sec^-1 (-2)`

Answer 1

Let `tan^-1(sqrt(3)) = α, "where"  (-pi)/(2) < α < pi/(2)`

∴ tan α = √3 = tan  `pi/(3)`

∴ α = `pi/(3)                    ...[∵ (-pi)/(2) < pi/(3) < pi/(2)]`

∴ `tan^-1(√3) = pi/(3)`        ...(1)

Let sec^-1(– 2) = β, where 0 ≤ β ≤ π, β ≠ `pi/(2)`

∴ sec β = – 2 = `- sec  (pi)/(3)`

∴ sec β = `sec(pi - pi/3)`   ...[∵ sec(π – θ) = – secθ]

∴ sec β = `sec  (2pi)/(3)`

∴ β = `(2pi)/(3)                 ...[∵ 0 ≤ (2pi)/(3) ≤ pi]`

∴ sec^{– 1}(– 2) = `(2pi)/(3)`              ...(2)

∴ `tan^-1(√3) - sec^-1(-2)`

= `pi/(3) - (2pi)/(3)`              ...[By (1) and (2)]

= `-(pi)/(3)`.