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Question
Find the principal value of the following:
`sec^-1 (-sqrt2)`
Solution
Let `sec^-1 (-sqrt2)` = y
`-sqrt2` = sec y
sec y = `- sqrt2`
`1/(cos y) = - sqrt2`
Taking reciprocal cos y = `((-1)/sqrt2)` [where 0 ≤ y ≤ π]
cos y = `cos (pi - pi/4) [cos pi/4 = 1/sqrt2 = cos (180^circ - theta) = - cos theta]`
`= cos ((4pi - pi)/4) = cos (3pi)/4`
∴ The principal value of sec-1 `(- sqrt2)` is `(3pi)/4`
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