LEARN · EXPLAIN · REVISE

Prove the identity by expanding both angles.

Read the idea, work independently, then explain what changed.

Return to the lesson / paper ↗

高一必修第一册(A版).pdf · 5.5 · PDF 222 / printed page 215

Revisit first: Unit-circle coordinates and signs

TOPIC 01

Identity choice and domain preservation

Transform expressions with a purpose, explain excluded angles and test tempting false rules.

What you will be able to explain

Choose an identity that matches the desired form.
Separate equality on a shared domain from extending a simplified formula.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

Examples 1–3: study a full demonstration; finish a solution from its first step; then choose a method independently before opening the solution. The remaining exercises may continue after class.

01 / Standard#Your turn

Prove the identity by expanding both angles.

\sin(x+y)+\sin(x-y)=2\sin x\cos y

Both angles use the same angular unit.

Skills and prerequisite lessons

Scope: JM01 / JM02

Working and explanation

BUILD THE REASONING

Hint 1

Use the addition and subtraction formulas.

Hint 2

The cos x sin y terms cancel.

Worked solution

Expand both terms with correct signs.
$\sin x\cos y+\cos x\sin y+\sin x\cos y-\cos x\sin y$
Collect identical surviving terms.
$2\sin x\cos y$

The identity holds for all real angles in consistent units.

Checks and common pitfalls: No division is used, so no extra exclusions are introduced.

Reasoning checklist · self / teacher assessment

Expand both terms with correct signs.
Collect identical surviving terms.

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

Teacher preparation and assessment

40-minute lesson

Minutes	Focus	Teaching and student action
5	Test two claims	Contrast false additivity with a domain hole using diagnostics.
8	Choose a destination	Students name the target form before using an identity.
10	Track exclusions	Pairs simplify the double-angle quotient and mark all forbidden angles.
12	Prove a bound	Groups construct the complementary square for the amplitude bound.
5	Exit domain	Require formula plus exclusions, not just the simplified value.

Question sequence

Which feature of the target expression suggests this identity?
Does the operation preserve the domain?

Board development

Record restrictions before cancellation.
$\sin2x\ne0$
Turn an identity into a bound.
$u^2+v^2=25$

Expected correct and incorrect responses

“The values agree where the original denominator is nonzero” correctly limits equality.
“It simplifies to 1 everywhere” incorrectly fills holes.

Follow-up questions

How do we prove the bound is attainable rather than merely valid?

Remediation

For false additivity, evaluate one concrete angle pair before teaching the correct expansion.

Exit check

Simplify sin²x/(1−cos²x) with its domain. Expected 1 for x≠kπ.

Assessment checklist and sample responses

2 marks: justified identity choice.
Full: numerator becomes sin²x matching the denominator; partial: unexplained cancellation.
2 marks: domain and equality cases.
Full: x≠kπ/2 retained; partial: only cosine zeros excluded.