Trigonometric identity — Power reduction

Question

Evaluate .

This practice question was inspired by Question 11(c) in the 2025 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..
Simplify the previous line carefully, keeping exact values where possible.An antiderivative is .
Finish the calculation, then check that the result meets the question’s conditions.At and , the sine terms vanish, leaving .
State the final answer clearly in the form the question requested.Final answer:

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

Verification: The average value over this half-wave is , times interval length .

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