Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| Expression | Value | ||
| (A) | \(\sin^{-1}\left(\sin\displaystyle\frac{3\pi}{4}\right)\) | (I) | \(\displaystyle\frac{\pi}{4}\) |
| (B) | \(\cos^{-1}\left(\cos\left(-\displaystyle\frac{2\pi}{3}\right)\right)\) | (II) | \(\displaystyle\frac{3\pi}{4}\) |
| (C) | \(\tan^{-1}\left(\tan\displaystyle\frac{7\pi}{6}\right)\) | (III) | \(\displaystyle\frac{2\pi}{3}\) |
| (D) | \(\cot^{-1}\left(\cot\left(-\displaystyle\frac{\pi}{4}\right)\right)\) | (IV) | \(\displaystyle\frac{\pi}{6}\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A)-(I), (B)-(III), (C)-(IV), (D)-(II)
An inverse function returns only values in its principal value branch: \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) for \(\sin^{-1}\), \([0, \pi]\) for \(\cos^{-1}\), \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) for \(\tan^{-1}\) and \((0, \pi)\) for \(\cot^{-1}\). Each angle is rewritten as an angle in the right branch with the same trigonometric value.
(A) \(\sin^{-1}\left(\sin\frac{3\pi}{4}\right)\)
This matches (I) \(\frac{\pi}{4}\), since sine does not change when the angle is replaced by \(\pi\) minus the angle:
\( \displaystyle \sin\frac{3\pi}{4} = \sin\left(\pi - \frac{3\pi}{4}\right) = \sin\frac{\pi}{4} \)
(B) \(\cos^{-1}\left(\cos\left(-\frac{2\pi}{3}\right)\right)\)
This matches (III) \(\frac{2\pi}{3}\), since cosine is an even function and \(\frac{2\pi}{3}\) is in \([0, \pi]\):
\( \displaystyle \cos\left(-\frac{2\pi}{3}\right) = \cos\frac{2\pi}{3} \)
(C) \(\tan^{-1}\left(\tan\frac{7\pi}{6}\right)\)
This matches (IV) \(\frac{\pi}{6}\), since tangent has period \(\pi\):
\( \displaystyle \tan\frac{7\pi}{6} = \tan\left(\frac{7\pi}{6} - \pi\right) = \tan\frac{\pi}{6} \)
(D) \(\cot^{-1}\left(\cot\left(-\frac{\pi}{4}\right)\right)\)
This matches (II) \(\frac{3\pi}{4}\), since cotangent has period \(\pi\) and \(\frac{3\pi}{4}\) is in \((0, \pi)\):
\( \displaystyle \cot\left(-\frac{\pi}{4}\right) = \cot\left(-\frac{\pi}{4} + \pi\right) = \cot\frac{3\pi}{4} \)