Question

Evaluate the following:

`"cosec"^-1(-sqrt(2)) + cot^-1(sqrt(3))`

Answer 1

Let `"cosec"^-1(-sqrt(2)) = α, "where"  (-pi)/(2) ≤ y ≤ pi/(2), y ≠ 0`

∴ cosec α = `- sqrt(2) = -"cosec"  pi/(4)`

∴ cosec α = `"cosec"(-pi/4)`   ...[∵ cosec (– θ) = – cosec θ]]

∴ α = `- pi/(4)                    ...[∵ (-pi)/(2) ≤ (-pi)/(4) ≤ pi/(2)]`

∴ `"cosec"^-1(- sqrt(2)) = - pi/(4)`                            ...(1)

Let `cot^-1(sqrt(3))` = β, where 0 < β < π

∴ cot β = `sqrt(3) = cot  (pi)/(6)`

∴ β = `(pi)/(6)                   ...[∵ 0 < (pi)/(6) < pi]`

∴ `cot^-1(sqrt(3)) = (pi)/(6)`              ...(2)

∴ `"cosec"^-1(-sqrt(2)) + cot^-1(sqrt(3))`

= `-pi/(4) + (pi)/(6)`              ...[By (1) and (2)]

= `(-3pi + 2pi)/(12)`

= `-(pi)/(12)`.