Trigonometry & Inverse Trigonometry
Simplification of inverse trig expressions
nta_pyq_2023_jan
Grade None
Question:
\tan^{-1}\left(\frac{1+\sqrt{3}}{3+\sqrt{3}}\right) + \sec^{-1}\left(\sqrt{\frac{8+4\sqrt{3}}{6+3\sqrt{3}}}\right) \text{ is equal to}
\frac{\pi}{4}
\frac{\pi}{2}
\frac{\pi}{3}
\frac{\pi}{6}
Step-by-Step Solution
Key Concept: Simplify each term by factoring; use \tan^{-1}(1/\sqrt{3}) = \pi/6 and \sec^{-1}(2/\sqrt{3}) = \pi/6, then add.
Simplify: \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) + \sec^{-1}\left(\frac{2}{\sqrt{3}}\right) = \frac{\pi}{6} + \frac{\pi}{6} = \frac{\pi}{3}
Correct Answer: 3