The value of \(\cos^{-1}\left(\sin\left(\cos^{-1}\displaystyle\frac{1}{2}\right)\right) + \tan^{-1}(1)\)
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{5\pi}{12}\)
Working from the inside, using principal values:
\( \displaystyle \cos^{-1}\frac{1}{2} = \frac{\pi}{3} \)
\( \displaystyle \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2} \)
\( \displaystyle \cos^{-1}\frac{\sqrt{3}}{2} = \frac{\pi}{6} \)
\( \displaystyle \tan^{-1}(1) = \frac{\pi}{4} \)
Adding:
\( \displaystyle \frac{\pi}{6} + \frac{\pi}{4} = \frac{2\pi + 3\pi}{12} = \frac{5\pi}{12} \)