Which of the following statements are true?
I. $\tan^{-1}(-x) = -\tan^{-1} x$ for every real $x$ .
II. $\cot^{-1}(-x) = -\cot^{-1} x$ for every real $x$ .
III. $\sec^{-1}(-x) = \pi - \sec^{-1} x$ for $\lvert x \rvert \ge 1$ .
IV. $\operatorname{cosec}^{-1}(-x) = \pi - \operatorname{cosec}^{-1} x$ for $\lvert x \rvert \ge 1$ .

I, III and IV onlyCorrect answerYour answerI, II and III onlyCorrect answerYour answerI and III onlyCorrect answerYour answerI and II onlyCorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer C

Correct answer: C

$\tan^{-1}$ and $\operatorname{cosec}^{-1}$ have branches symmetric about 0, so a negative input just flips the sign: I is true, IV is false ( $\operatorname{cosec}^{-1}(-1) = -\frac{\pi}{2}$ , not $\frac{\pi}{2}$ ).
$\cot^{-1}$ and $\sec^{-1}$ have branches inside $[0, \pi]$ , so a negative input gives $\pi$ minus the angle: III is true, II is false ( $\cot^{-1}(-1) = \frac{3\pi}{4}$ , not $-\frac{\pi}{4}$ ).

Worth noting

IV gives $\operatorname{cosec}^{-1}$ the rule of $\sec^{-1}$ ; its branch is the one of $\sin^{-1}$ , around 0.

Concept: Principal values of related inputs
Typical timeTypical: 150 s
Your timeYours: —

The rule this testsPrincipal values

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Ten questions from Inverse Trigonometric Functions, at your own pace.

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