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प्रश्न
Evaluate the following:
`tan^-1 sqrt(3) - sec^-1 (-2)`
उत्तर
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)`.
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