Identities are transformation tools
An identity is true for every angle where both sides are defined. It lets us replace one expression by an equivalent form; the task is to recognise which replacement simplifies the structure.
Learning goals
- Distinguish an identity from an equation true only for selected angles.
- Derive and recall the three Pythagorean identities.
- Select the identity matching an expression's structure.
- Convert expressions using reciprocal and quotient identities.
Concept lesson
Reciprocal and quotient relations
cosec=1/sin, sec=1/cos, cot=1/tan, tan=sin/cos and cot=cos/sin. These relations can convert an expression to one function family.
Pythagoras divided three ways
Start with O squared plus A squared equals H squared. Divide by H squared, A squared or O squared to obtain the three identity families.
Identity versus conditional equation
sin squared A plus cos squared A equals 1 is an identity. sin A equals cos A is true only for selected A and is an equation to solve under a stated domain. Testing one angle can disprove a proposed identity but cannot prove it universally.
The three Pythagorean identities
Dividing by H squared gives sin squared A plus cos squared A equals 1. Dividing by A squared gives 1 plus tan squared A equals sec squared A. Dividing by O squared gives 1 plus cot squared A equals cosec squared A. Rearrange rather than memorise sign variants separately.
Match the visible pair to an identity
sec squared A minus tan squared A becomes 1 by rearranging 1 plus tan squared A equals sec squared A. cosec squared A minus cot squared A similarly becomes 1. Name the parent identity before rearranging.
Convert to sin and cos
For tan A times sin A plus cos A, replace tan A by sin A/cos A, combine over cos A and use sin squared A plus cos squared A equals 1 to obtain sec A. The conversion creates the identity pattern deliberately.
Identity, equation or false statement
Classify statements using definitions and counterexamples. For identities, indicate the transformation or parent identity; for equations, state that only selected angles may satisfy them.
Repair the identity family
Correct sign and function errors in proposed Pythagorean identities by returning to the appropriate divided form of Pythagoras.
Choose a productive conversion
Compare possible conversions and choose the one that reduces the number of function families or creates a Pythagorean pair. Reject conversions that only increase complexity.
Find an expression without finding the angle
Use a given relation such as sin A plus sin squared A equals 1 to substitute into sin squared A plus cos squared A equals 1 and determine the requested expression without solving for A.
One-mark identity recognition
Identify the shortest matching relation before calculating. Many one-mark expressions collapse immediately through a reciprocal product or a rearranged Pythagorean identity.
Identity tools mastery
Classify statements, derive all three Pythagorean identities, simplify direct variants, and solve two mixed-function expressions by selecting productive conversions.