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प्रश्न
`tan^(-1) sqrt3 - sec^(-1)(-2)` is equal to ______.
पर्याय
π
`-pi/3`
`pi/3`
`(2pi)/3`
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उत्तर
`tan^(-1) sqrt3 - sec^(-1)(-2)` is equal to `underlinebb(-pi/3)`.
Explanation:
We know that the range of principal value branch of tan−1 and sec−1 are `(-pi/2, pi/2)` and [0, π] respectively.
Let `tan^(-1) sqrt3` = x ⇒ `sqrt3` = tan x
Then `sqrt3 = tan(pi/3)`
Where `pi/3 ∈ (-pi/2, pi/2)`
Let sec−1 (−2) = y ⇒ −2 = sec y
Then, −2 = `sec(pi/3)`
= `sec(pi - pi/3) = sec (2pi)/3`
Where `(2pi)/3 ∈ [0, pi] - {pi/2}`
∴ `tan^(-1) sqrt3 - sec^(-1)(-2)`
= `pi/3 - (pi - pi/3)`
= `pi/3 - (2pi)/3`
= `-pi/3`
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