A proof is a reversible path
Start from one side, normally the more complex side, and transform it through valid equal expressions until it becomes the untouched target side. Each line must follow from the previous one.
Learning goals
- Choose the more complex side and preserve the other as target.
- Use factoring, common denominators and conjugates strategically.
- Convert to a productive ratio family and apply identities.
- Write a valid one-direction proof with equality preserved.
Concept lesson
Identity toolbox
Recall reciprocal and quotient relations plus sin squared plus cos squared equals 1, sec squared minus tan squared equals 1 and cosec squared minus cot squared equals 1.
Algebra moves used in proofs
Use a squared minus b squared=(a-b)(a+b), common denominators, factor cancellation and conjugate multiplication. Cancellation is legal only for a common nonzero factor of the entire numerator and denominator.
Plan from structure and target
Inspect both sides before writing. Choose the side with more operations or function families. Read the target for clues: a target sec suggests creating 1/cos; a target 1 suggests a reciprocal product or Pythagorean difference.
Cancellation requires factors
In a fraction, terms joined by plus or minus cannot be cancelled individually. First factor the whole numerator or denominator. For example, (1-sin squared A)/cos A becomes cos squared A/cos A only through the identity, then cancels the common cos factor.
Use a common denominator and factor
For cos A/(1+sin A)+(1+sin A)/cos A, combine over cos A(1+sin A), simplify the numerator using sin squared plus cos squared equals 1, factor 2(1+sin A), cancel the common factor and obtain 2sec A.
Create a Pythagorean difference
In sec A(1-sin A)(sec A+tan A), convert sec A(1-sin A) to sec A-tan A. The product becomes (sec A-tan A)(sec A+tan A)=sec squared A-tan squared A=1.
Convert all functions to sin and cos
When an expression mixes tan, cot, sec and cosec without an immediate conjugate, rewrite them in sin and cos, combine factors, create sin squared plus cos squared where possible and simplify toward the target.
Choose one starting side
For four identities, select the more complex starting side, identify the target form and complete a one-direction chain. Reject circular proofs that begin by assuming the required equality.
Legal or illegal cancellation
Classify proposed cancellation steps, factor expressions where possible and rewrite illegal term cancellation as a valid identity or common-factor step.
Create the identity before using it
For each expression, identify the desired Pythagorean pair, perform quotient or reciprocal conversions that create it, then apply the identity with the matching terms visible.
Compare two valid proof routes
Complete a conjugate route and an all-sin-cos route for the same identity. Compare their number of transformations and select the clearer examination solution without treating the alternative as invalid.
Three-mark proof checklist
Start with LHS or the more complex side, show every denominator/factor step, cite or visibly apply identities, avoid illegal cancellation, reach the untouched RHS and close with Hence proved.
Identity-proof mastery
Prove one common-denominator identity, one conjugate identity and one mixed-function identity. Audit a fourth flawed proof for circular reasoning, illegal cancellation and unmatched identity use.