The value of \(\tan\left(\sin^{-1}\left(-\displaystyle\frac{5}{13}\right)\right)\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(-\displaystyle\frac{5}{12}\)
Let \(\theta = \sin^{-1}\left(-\frac{5}{13}\right)\). Then
\( \displaystyle \sin\theta = -\frac{5}{13} \)
Since the principal value branch is \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) and \(\sin\theta\) is negative, \(\theta\) lies in \(\left[-\frac{\pi}{2}, 0\right)\), where the cosine is positive:
\( \displaystyle \cos\theta = \sqrt{1 - \frac{25}{169}} \)
\( \displaystyle = \sqrt{\frac{144}{169}} = \frac{12}{13} \)
Hence
\( \displaystyle \tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{-5/13}{12/13} = -\frac{5}{12} \)
Why the other options are incorrect
Option (1) \(\frac{5}{12}\) — Incorrect. \(\theta\) is negative, so \(\tan\theta\) is negative.
Option (2) \(-\frac{12}{5}\) — Incorrect. This is \(\cot\theta\), not \(\tan\theta\).
Option (4) \(\frac{12}{13}\) — Incorrect. This is \(\cos\theta\), not \(\tan\theta\).