Which of the following statements are correct? (A) sin^-1(-x)=-sin^-1…

The correct answer is Option (3) → (A), (C) and (E) Only

The rule for \(f^{-1}(-x)\) depends on the principal value branch:

    \(\sin^{-1}\): \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\), symmetric about 0
    \(\tan^{-1}\): \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), symmetric about 0
    \(\cos^{-1}\): \([0, \pi]\)
    \(\sec^{-1}\): \([0, \pi] - \left\{\frac{\pi}{2}\right\}\)
    \(\cot^{-1}\): \((0, \pi)\)

When the branch is symmetric about 0, the function is odd: \(f^{-1}(-x) = -f^{-1}(x)\). When the branch lies in \([0, \pi]\), the rule is \(f^{-1}(-x) = \pi - f^{-1}(x)\).

So \(\sin^{-1}\) and \(\tan^{-1}\) follow \(f^{-1}(-x) = -f^{-1}(x)\), while \(\cos^{-1}\), \(\sec^{-1}\) and \(\cot^{-1}\) follow \(f^{-1}(-x) = \pi - f^{-1}(x)\).

Checking the statements

(A) \(\sin^{-1}(-x)=-\sin^{-1}x,\ x \in[-1, 1]\) — Correct. \(\sin^{-1}\) is odd on its whole domain.

(B) \(\tan^{-1}(-x)=\pi-\tan^{-1}x,\ -1 \lt x \lt 1\) — Incorrect. At x = 0 the left side is 0 but the right side is π; the true rule is \(\tan^{-1}(-x) = -\tan^{-1}x\).

(C) \(\cos^{-1}(-x)=\pi-\cos^{-1}x,\ -1 \le x \le +1\) — Correct. The branch of \(\cos^{-1}\) is \([0, \pi]\).

(D) \(\sec^{-1}(-x)=-\sec^{-1}x,\ |x| \gt 1\) — Incorrect. \(\sec^{-1}(-2) = \frac{2\pi}{3}\), but \(-\sec^{-1}2 = -\frac{\pi}{3}\); the true rule is \(\pi - \sec^{-1}x\).

(E) \(\cot^{-1}(-x)=\pi-\cot^{-1}x,\ x \in R\) — Correct. The branch of \(\cot^{-1}\) is \((0, \pi)\).

Note: This topic was not in the CUET 2026 syllabus.