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Trig identities

15 formulas · what each one means · a free lesson for each

All formula sheets

steplyclass.com/formula-sheets/trig-identities/ · free, step-by-step lessons for every formula

Basic identities

Reciprocal
csc θ = 1sin θ sec θ = 1cos θ cot θ = 1tan θ
Lesson: Trig identities →
Quotient
tan θ = sin θcos θ cot θ = cos θsin θ
Lesson: Trig identities →
Pythagorean
sin2 θ + cos2 θ = 1
Pythagoras on the unit circle.Lesson: Trig identities →
Pythagorean (tan)
1 + tan2 θ = sec2 θ
Divide the first one by cos²θ.Lesson: Trig identities →
Pythagorean (cot)
1 + cot2 θ = csc2 θ
Divide the first one by sin²θ.Lesson: Trig identities →
Even and odd
cos(−θ) = cos θ sin(−θ) = −sin θ tan(−θ) = −tan θ
Lesson: Trig identities →
Cofunction
sin θ = cos(90° − θ) tan θ = cot(90° − θ)
Lesson: sin of one angle = cos of the other →

Sum and difference identities

Sine
sin(A ± B) = sin A cos B ± cos A sin B
Lesson: Sum and difference identities →
Cosine
cos(A ± B) = cos A cos B ∓ sin A sin B
The sign turns over.Lesson: Sum and difference identities →
Tangent
tan(A ± B) = tan A ± tan B1 ∓ tan A tan B
Lesson: Sum and difference identities →

Double-angle identities

Sine
sin 2θ = 2 sin θ cos θ
Lesson: Double-angle identities →
Cosine
cos 2θ = cos2 θ − sin2 θ = 2cos2 θ − 1 = 1 − 2sin2 θ
Three forms: pick the one with what you know.Lesson: Double-angle identities →
Tangent
tan 2θ = 2 tan θ1 − tan2 θ
Lesson: Double-angle identities →

Power-reducing (half-angle) identities

sin²
sin2 x = 1 − cos 2x2
Used to integrate even powers of sin.Lesson: Trig integrals →
cos²
cos2 x = 1 + cos 2x2
Used to integrate even powers of cos.Lesson: Trig integrals →

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