Q106 / 180 · COMEDK 2024 (Evening Shift) · Mathematics
$\text { Evaluate: } \cot ^{-1}\left(-\dfrac{3}{\sqrt{3}}\right)-\sec ^{-1}\left(-\dfrac{2}{\sqrt{2}}\right)-\operatorname{cosec}^{-1}(-1)-\tan ^{-1}(1)$
- A$-\dfrac{2 \pi}{3}$
- B0
- Correct answerC$\dfrac{\pi}{3}$
- D$\dfrac{\pi}{6}$
Solution
The given expression is $E = \cot ^{-1}\left(-\dfrac{3}{\sqrt{3}}\right) - \sec ^{-1}\left(-\dfrac{2}{\sqrt{2}}\right) - \operatorname{cosec}^{-1}(-1) - \tan ^{-1}(1)$.
Simplify the arguments:
$\dfrac{3}{\sqrt{3}} = \sqrt{3}$ and $\dfrac{2}{\sqrt{2}} = \sqrt{2}$.
Evaluate each term individually:
1. $\cot ^{-1}(-\sqrt{3}) = \pi - \cot ^{-1}(\sqrt{3}) = \pi - \dfrac{\pi}{6} = \dfrac{5\pi}{6}$.
2. $\sec ^{-1}(-\sqrt{2}) = \pi - \sec ^{-1}(\sqrt{2}) = \pi - \dfrac{\pi}{4} = \dfrac{3\pi}{4}$.
3. $\operatorname{cosec}^{-1}(-1) = -\operatorname{cosec}^{-1}(1) = -\dfrac{\pi}{2}$.
4. $\tan ^{-1}(1) = \dfrac{\pi}{4}$.
Substitute these values into the expression $E$:
$E = \dfrac{5\pi}{6} - \dfrac{3\pi}{4} - \left(-\dfrac{\pi}{2}\right) - \dfrac{\pi}{4}$.
Combine the terms:
$E = \dfrac{5\pi}{6} - \dfrac{3\pi}{4} + \dfrac{\pi}{2} - \dfrac{\pi}{4} = \dfrac{5\pi}{6} - \pi + \dfrac{\pi}{2} = \dfrac{5\pi - 6\pi + 3\pi}{6} = \dfrac{2\pi}{6} = \dfrac{\pi}{3}$.
Answer: $\dfrac{\pi}{3}$
Simplify the arguments:
$\dfrac{3}{\sqrt{3}} = \sqrt{3}$ and $\dfrac{2}{\sqrt{2}} = \sqrt{2}$.
Evaluate each term individually:
1. $\cot ^{-1}(-\sqrt{3}) = \pi - \cot ^{-1}(\sqrt{3}) = \pi - \dfrac{\pi}{6} = \dfrac{5\pi}{6}$.
2. $\sec ^{-1}(-\sqrt{2}) = \pi - \sec ^{-1}(\sqrt{2}) = \pi - \dfrac{\pi}{4} = \dfrac{3\pi}{4}$.
3. $\operatorname{cosec}^{-1}(-1) = -\operatorname{cosec}^{-1}(1) = -\dfrac{\pi}{2}$.
4. $\tan ^{-1}(1) = \dfrac{\pi}{4}$.
Substitute these values into the expression $E$:
$E = \dfrac{5\pi}{6} - \dfrac{3\pi}{4} - \left(-\dfrac{\pi}{2}\right) - \dfrac{\pi}{4}$.
Combine the terms:
$E = \dfrac{5\pi}{6} - \dfrac{3\pi}{4} + \dfrac{\pi}{2} - \dfrac{\pi}{4} = \dfrac{5\pi}{6} - \pi + \dfrac{\pi}{2} = \dfrac{5\pi - 6\pi + 3\pi}{6} = \dfrac{2\pi}{6} = \dfrac{\pi}{3}$.
Answer: $\dfrac{\pi}{3}$
Hence, the correct answer is option: 'C'.