Trigonometric identity — Cosine with half RHS

Question

(i) Show that .

(ii) By letting in the cubic equation , show that .

(iii) Prove that .

This practice question was inspired by Question 14(b) in the 2020 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: apply the named identity first and simplify before integrating or solving.(i) Expand and simplify using double-angle formulae to obtain .
Simplify the previous line carefully, keeping exact values where possible.(ii) Substitute : . Divide by : .
Simplify the previous line carefully, keeping exact values where possible.From (i), , so .
Simplify the previous line carefully, keeping exact values where possible.(iii) From , angles give three distinct roots of .
Finish the calculation, then check that the result meets the question’s conditions.Vieta: sum , sum of products two at a time , so .
State the final answer clearly in the form the question requested.Hence .

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

Verification: Check angles: , , , all with .

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