Question

find the principal value of tan¹ (-1).​

Answer 1

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let y = tan^-1(-1)

tan y = 1

tan y = tan(-π/4)

range of principal value of tan^-1 is

(-π/2 to π/2)

so principal value of tan^-1(-1) is -π/4

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Answer 2

Step-by-step explanation:

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Let $\sf tan^{-1}(1)\:=\:y$ ...Then,

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$\sf tan\,y\:=\:-1\:=\:-tan \left(\dfrac{\pi}{4}\right)\:=\:tan \left(-\dfrac{\pi}{4}\right)$

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We know that the range of the principle value of $\sf tan^{-1}$ is

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$\sf \left(- \dfrac{\pi}{2}\:,\: \dfrac{\pi}{2}\right)$ and $\sf tan \left(-\dfrac{\pi}{4}\right)$ is -1

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Therefore, principle value of $\sf tan^{-1}$ is $\tt -\dfrac{\pi}{4}$