← Senior Mathematics Studio

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Is the original expression defined at x=0?

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

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

Revisit first: Unit-circle coordinates and signs

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.