Evaluate: cot-1(1/√3) − cosec-1(−2) + sec-1(2/√3)
Solution:
Step 1: Evaluate cot⁻¹(1/√3)
\[ \cot^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{3} \]
Step 2: Evaluate cosec⁻¹(−2)
\[ \csc^{-1}(-2) = -\frac{\pi}{6} \]
Step 3: Evaluate sec⁻¹(2/√3)
\[ \sec^{-1}\left(\frac{2}{\sqrt{3}}\right) = \frac{\pi}{6} \]
Step 4: Substitute
\[ \frac{\pi}{3} – \left(-\frac{\pi}{6}\right) + \frac{\pi}{6} \]
\[ = \frac{\pi}{3} + \frac{\pi}{6} + \frac{\pi}{6} = \frac{\pi}{3} + \frac{\pi}{3} = \frac{2\pi}{3} \]
Final Answer:
Value = \[ \frac{2\pi}{3} \]