Odd/Even Identities
High School
Definition
A way of simplifying and grouping different functions. We can define trigonometry identities as either a odd or even function. For example, sin and tan are odd functions while cos is an even function.
Worked examples
\(\sin(-x) = -\sin(x)\)
Sine is odd: flipping the sign of the input flips the sign of the output.
\(\cos(-x) = \cos(x)\)
Cosine is even: flipping the sign of the input leaves the output unchanged.
\(\tan(-x) = -\tan(x)\)
Tangent is odd, just like sine; the negative moves through the function.
Common mistakes
- \(\cos(-x) = -\cos(x)\) → \(\cos(-x) = \cos(x)\) Cosine is even, not odd; the output stays positive when the input flips sign.
- \(\sin(-x) = \sin(x)\) → \sin(-x) = -\sin(x)\) Sine is odd; the negative sign must carry through to the output.
- All trig functions are odd. → Only sine, tangent, cosecant, and cotangent are odd; cosine and secant are even. Cosine and secant are even functions; don't overgeneralize the odd property.
Where you'll use it next
You'll apply odd/even identities when simplifying trig expressions, solving equations with negative angles, proving more complex identities, and analyzing symmetry in precalculus and calculus.
Found in 1 StudyPug lesson
Even and Odd Functions: Mastering Identification and Application
12th Grade12thGrade 12 Math
Unlock the power of even and odd functions! Learn foolproof methods to identify, analyze, and apply these crucial mathematical concepts. Boost your problem-solving skills and excel in algebra and calculus.
See also
Odd functionRatio IdentitiesHalf Number IdentitiesIdentity (Equation)Even NumbersTangent (in Trigonometry)
Reviewed by Pat Cheng, M.Ed. — StudyPug Curriculum Lead · Last updated June 6, 2026