Inverse trigonometric functions — Different right-hand side

Question

(i) Explain briefly why has at most one solution for . Hence,

(ii) Solve .

This practice question was inspired by Question 14(c) in the 2024 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Set up the solution: use the restricted domain and range that make the inverse function single-valued.(i) Sum is strictly increasing (sum of strictly increasing continuous functions).
Finish the calculation, then check that the result meets the question’s conditions.Using part (i), (ii) . Roots , ; reject (sum negative / undefined issues with product of args).
State the final answer clearly in the form the question requested.Answer: .

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: Same tan-addition and root-checking pattern.

Any mark labels are a TestMum study aid, not an official NESA marking allocation.