Evaluate cos⁻¹(−1/2) + 2 sin⁻¹(−1/2) − tan⁻¹(−1).

1. 1 · Write what is given and what is to be found. To find: the value of cos⁻¹(−1/2) + 2 sin⁻¹(−1/2) − tan⁻¹(−1), using principal values.
2. 2 · Write the principal branches. cos⁻¹: [0, π]. sin⁻¹: [−π/2, π/2]. tan⁻¹: (−π/2, π/2).
3. 3 · First term. cos(π/3) = 1/2, so cos⁻¹(−1/2) = π − π/3 = 2π/3.
4. 4 · Second term. sin(π/6) = 1/2, so sin⁻¹(−1/2) = −π/6, and 2 sin⁻¹(−1/2) = −π/3.
5. 5 · Third term. tan(π/4) = 1, so tan⁻¹(−1) = −π/4.
6. 6 · Combine. 2π/3 − π/3 − (−π/4) = π/3 + π/4 = 4π/12 + 3π/12 = 7π/12.
7. 7 · Check and box the answer. Each value lies in its own branch. The answer is 7π/12.

Scoring tip: Write each principal value on its own line before you combine them. A sign slip in one term is then easy to spot.