Trigonometric identity — Cosine triple-angle chain

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 .
Simplify the previous line carefully, keeping exact values where possible.Use and to obtain .
Simplify the previous line carefully, keeping exact values where possible.Rearrange: , then divide by to get the required form.
Simplify the previous line carefully, keeping exact values where possible.(ii) Substitute : .
Simplify the previous line carefully, keeping exact values where possible.Divide by : . From (i), , so .
Simplify the previous line carefully, keeping exact values where possible.(iii) From , the angles give three distinct roots of .
Simplify the previous line carefully, keeping exact values where possible.By Vieta, sum of roots and sum of products two at a time .
Finish the calculation, then check that the result meets the question’s conditions.Hence .
State the final answer clearly in the form the question requested.But , so the cosine sum is .

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

Verification: Identity ; cubic has no term so .

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