The value of \(\sec\left(\tan^{-1}\left(-\displaystyle\frac{3}{4}\right)\right)\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{5}{4}\)
Let \(\theta = \tan^{-1}\left(-\frac{3}{4}\right)\). Then \(\tan\theta = -\frac{3}{4}\), and \(\theta\) lies in the principal value branch \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\). Since \(\tan\theta\) is negative,
\( \displaystyle -\frac{\pi}{2} \lt \theta \lt 0 \)
In this fourth-quadrant interval the cosine, and so the secant, is positive.
Using \(\sec^{2}\theta = 1 + \tan^{2}\theta\):
\( \displaystyle \sec^{2}\theta = 1 + \frac{9}{16} = \frac{25}{16} \)
Taking the positive root:
\( \displaystyle \sec\theta = \frac{5}{4} \)
Why the other options are incorrect
Option (1) \(-\frac{5}{4}\) — Incorrect. The angle is between \(-\frac{\pi}{2}\) and 0, where the secant is positive.
Option (2) \(\frac{5}{3}\) — Incorrect. \(\frac{5}{3}\) is the size of the cosecant, not the secant; in fact \(\mathrm{cosec}\,\theta = -\frac{5}{3}\).
Option (4) \(-\frac{5}{3}\) — Incorrect. This is \(\mathrm{cosec}\,\theta\), not \(\sec\theta\).