Q63 / 180 · COMEDK 2024 (Afternoon Shift) · Mathematics
Evaluate :
$ \operatorname{cosec}^{-1}\left(-\dfrac{2 \sqrt{3}}{3}\right)+\tan ^{-1}\left(-\dfrac{\sqrt{3}}{3}\right)+\sec ^{-1} 2+\cos ^{-1}\left(-\dfrac{1}{2}\right)-\sin ^{-1}\left(\dfrac{\sqrt{2}}{2}\right)$
- A$-\dfrac{\pi}{12}$
- B$\dfrac{5 \pi}{4}$
- C$-\dfrac{5\pi}{12}$
- Correct answerD$\dfrac{\pi}{4}$
Solution
The expression is $E = \operatorname{cosec}^{-1}\left(-\dfrac{2\sqrt{3}}{3}\right) + \tan^{-1}\left(-\dfrac{\sqrt{3}}{3}\right) + \sec^{-1}(2) + \cos^{-1}\left(-\dfrac{1}{2}\right) - \sin^{-1}\left(\dfrac{\sqrt{2}}{2}\right)$.
Simplify each term:
1. $\operatorname{cosec}^{-1}\left(-\dfrac{2}{\sqrt{3}}\right) = -\operatorname{cosec}^{-1}\left(\dfrac{2}{\sqrt{3}}\right) = -\dfrac{\pi}{3}$.
2. $\tan^{-1}\left(-\dfrac{1}{\sqrt{3}}\right) = -\tan^{-1}\left(\dfrac{1}{\sqrt{3}}\right) = -\dfrac{\pi}{6}$.
3. $\sec^{-1}(2) = \dfrac{\pi}{3}$.
4. $\cos^{-1}\left(-\dfrac{1}{2}\right) = \pi - \cos^{-1}\left(\dfrac{1}{2}\right) = \pi - \dfrac{\pi}{3} = \dfrac{2\pi}{3}$.
5. $\sin^{-1}\left(\dfrac{1}{\sqrt{2}}\right) = \dfrac{\pi}{4}$.
Summing these values:
$E = -\dfrac{\pi}{3} - \dfrac{\pi}{6} + \dfrac{\pi}{3} + \dfrac{2\pi}{3} - \dfrac{\pi}{4}$
$E = -\dfrac{\pi}{6} + \dfrac{2\pi}{3} - \dfrac{\pi}{4}$
Finding a common denominator of 12:
$E = \dfrac{-2\pi + 8\pi - 3\pi}{12} = \dfrac{3\pi}{12} = \dfrac{\pi}{4}$.
Answer: $\dfrac{\pi}{4}$
Simplify each term:
1. $\operatorname{cosec}^{-1}\left(-\dfrac{2}{\sqrt{3}}\right) = -\operatorname{cosec}^{-1}\left(\dfrac{2}{\sqrt{3}}\right) = -\dfrac{\pi}{3}$.
2. $\tan^{-1}\left(-\dfrac{1}{\sqrt{3}}\right) = -\tan^{-1}\left(\dfrac{1}{\sqrt{3}}\right) = -\dfrac{\pi}{6}$.
3. $\sec^{-1}(2) = \dfrac{\pi}{3}$.
4. $\cos^{-1}\left(-\dfrac{1}{2}\right) = \pi - \cos^{-1}\left(\dfrac{1}{2}\right) = \pi - \dfrac{\pi}{3} = \dfrac{2\pi}{3}$.
5. $\sin^{-1}\left(\dfrac{1}{\sqrt{2}}\right) = \dfrac{\pi}{4}$.
Summing these values:
$E = -\dfrac{\pi}{3} - \dfrac{\pi}{6} + \dfrac{\pi}{3} + \dfrac{2\pi}{3} - \dfrac{\pi}{4}$
$E = -\dfrac{\pi}{6} + \dfrac{2\pi}{3} - \dfrac{\pi}{4}$
Finding a common denominator of 12:
$E = \dfrac{-2\pi + 8\pi - 3\pi}{12} = \dfrac{3\pi}{12} = \dfrac{\pi}{4}$.
Answer: $\dfrac{\pi}{4}$
Hence, the correct answer is option: 'D'.