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Question
Evaluate the following:
`"cosec"^-1(-sqrt(2)) + cot^-1(sqrt(3))`
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Solution
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)`.
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