Reciprocal identities and one-sided proof chains
| English | 中文 | Pinyin |
|---|---|---|
| identity/aɪˈdentɪti/ | 恒等式 | héng děng shì |
Two formulas agree at one angle. Does that prove they agree for every angle where they are defined?
- Two formulas agree at one angle. Does that prove they agree for every angle where they are defined?
- This lesson studies identity 恒等式: An equality true for every input in its stated common domain.
Choose the mathematical structure
- The unit-circle equation gives sin²θ+cos²θ=1. Divide by cos²θ when cosθ≠0 to obtain tan²θ+1=sec²θ. Divide by sin²θ when sinθ≠0 to obtain 1+cot²θ=cosec²θ. Record exclusions before cancelling or dividing. A proof starts from one side and reaches the other through valid equalities; numerical checks alone are not a proof.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines identity?
An equality true for every input in its stated common domain.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Prove tan²θ−sin²θ=tan²θ sin²θ for cosθ≠0. The left side is sin²θ(1/cos²θ−1)=sin²θ(1−cos²θ)/cos²θ=sin⁴θ/cos²θ=tan²θ sin²θ. No division by sinθ is needed, so sinθ=0 remains allowed. For sinθ≠0, (1−cosθ)/sinθ becomes (1−cosθ)(1+cosθ)/[sinθ(1+cosθ)]=sin²θ/[sinθ(1+cosθ)]=sinθ/(1+cosθ). The stated exclusion also makes 1+cosθ nonzero. If secθ=2 with θ acute, tan²θ=4−1=3, cosθ=1/2, sin²θ=3/4, cosec²θ=4/3 and cot²θ=1/3.
Reciprocal identities and one-sided proof chains
The unit-circle equation gives sin²θ+cos²θ=1
Explain which denominator or branch restriction makes each step valid.
If secθ=2 and θ is acute, find tan²θ.
tan²θ=sec²θ−1=4−1=3.
Test a tempting shortcut
- Proving an identity does not mean assuming both sides equal and manipulating that assumption without justification. A squared identity does not fix a function’s sign: the quadrant does. The last expression sinθ/(1+cosθ) is defined at θ=0, but the original (1−cosθ)/sinθ is not; simplification does not repair the original domain.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Checking a proposed identity at one angle proves it for all allowed angles. This claim is false. Explain which definition or assumption it violates.
Find cosec²θ for that angle.
sin²θ=1−cos²θ=3/4; its reciprocal is 4/3.
Checking a proposed identity at one angle proves it for all allowed angles.
Proving an identity does not mean assuming both sides equal and manipulating that assumption without justification. A squared identity does not fix a function’s sign: the quadrant does. The last expression sinθ/(1+cosθ) is defined at θ=0, but the original (1−cosθ)/sinθ is not; simplification does not repair the original domain.
Interpret a new situation
- Choose the more complicated side, replace reciprocal functions by sine/cosine, use a common denominator and apply sin²+cos²=1. Annotate every cancelled factor. To disprove a claimed identity, one valid counterexample suffices; to prove it, explain why the chain works for all angles in the common domain.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find cot²θ for that angle.
cot²θ=cosec²θ−1=4/3−1=1/3.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · E. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
An equality true for every input in its stated common domain. Choose the relationship, show the method, check its assumptions and interpret the result.