Derive the two difference-to-product identities from the angle addition and subtraction formulae.

01 / Standard#Your turn

\sin x-\sin y=2\cos\frac{x+y}2\sin\frac{x-y}2,\quad\cos x-\cos y=-2\sin\frac{x+y}2\sin\frac{x-y}2

Official paper · jm02-2023 · 5(a)(i) · PDF 7

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Set u=(x+y)/2 and v=(x−y)/2.

Hint 2

Subtract the expansions at u+v and u−v.

Worked solution

For sine, the sin u cos v terms cancel.
$\sin(u+v)-\sin(u-v)=2\cos u\sin v$
For cosine, the cos u cos v terms cancel. Substitute u,v back.
$\cos(u+v)-\cos(u-v)=-2\sin u\sin v$

Both identities follow by addition-formula cancellation.

Checks and common pitfalls: The cosine difference identity has a leading minus sign.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
For sine, the sin u cos v terms cancel.
$\sin(u+v)-\sin(u-v)=2\cos u\sin v$
For cosine, the cos u cos v terms cancel. Substitute u,v back.
$\cos(u+v)-\cos(u-v)=-2\sin u\sin v$

Think first. Reveal a hint when the class is ready.

2023 JM02