Evaluate: tan-1(−1/√3) + cot-1(1/√3) + tan-1(sin(−π/2))
Solution:
Step 1: Evaluate tan⁻¹(−1/√3)
\[ \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) = -\frac{\pi}{6} \]
Step 2: Evaluate cot⁻¹(1/√3)
\[ \cot^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{3} \]
Step 3: Evaluate sin(−π/2)
\[ \sin\left(-\frac{\pi}{2}\right) = -1 \]
Step 4: Evaluate tan⁻¹(−1)
\[ \tan^{-1}(-1) = -\frac{\pi}{4} \]
Step 5: Add all values
\[ -\frac{\pi}{6} + \frac{\pi}{3} – \frac{\pi}{4} \]
LCM = 12:
\[ = -\frac{2\pi}{12} + \frac{4\pi}{12} – \frac{3\pi}{12} = -\frac{\pi}{12} \]
Final Answer:
Value = \[ -\frac{\pi}{12} \]