Evaluate: tan-1(√3) − sec-1(−2)
Solution:
Step 1: Evaluate tan⁻¹(√3)
\[ \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \]
(Principal range: \((- \pi/2, \pi/2)\))
Step 2: Evaluate sec⁻¹(−2)
\[ \sec^{-1}(-2) = \frac{2\pi}{3} \]
(Since sec θ = −2 ⇒ cos θ = −1/2, and θ ∈ [0,π], θ ≠ π/2)
Step 3: Subtract
\[ \frac{\pi}{3} – \frac{2\pi}{3} = -\frac{\pi}{3} \]
Final Answer:
Value = \[ -\frac{\pi}{3} \]