Evaluate: cosec-1(−2/√3) + 2cot-1(−1)
Solution:
Step 1: Evaluate cosec⁻¹(−2/√3)
\[ \csc^{-1}\left(-\frac{2}{\sqrt{3}}\right) \Rightarrow \sin y = -\frac{\sqrt{3}}{2} \]
\[ y = -\frac{\pi}{3} \]
Step 2: Evaluate cot⁻¹(−1)
\[ \cot^{-1}(-1) \]
We know:
\[ \cot\left(\frac{3\pi}{4}\right) = -1 \]
\[ \Rightarrow \cot^{-1}(-1) = \frac{3\pi}{4} \]
Step 3: Substitute
\[ -\frac{\pi}{3} + 2\left(\frac{3\pi}{4}\right) \]
\[ = -\frac{\pi}{3} + \frac{3\pi}{2} \]
LCM = 6:
\[ = -\frac{2\pi}{6} + \frac{9\pi}{6} = \frac{7\pi}{6} \]
Final Answer:
Value = \[ \frac{7\pi}{6} \]