Trigonometry
Common Trigonometric Identities
A trigonometric identity is an equation that is true for every angle where both sides are defined. Identities rewrite expressions without changing their values.
Reviewed
Reciprocal and quotient identities
Secant, cosecant, and cotangent are reciprocals of cosine, sine, and tangent. Quotient identities express tangent and cotangent using sine and cosine.
Domain restrictions travel with the identity. For example, sec(theta) is undefined where cos(theta)=0.
sec=1/cos; csc=1/sin; cot=1/tan; tan=sin/cosThe Pythagorean identities
A unit-circle point satisfies x^2+y^2=1. Substituting x=cos(theta) and y=sin(theta) gives the fundamental identity.
Dividing that identity by cos^2(theta) or sin^2(theta) produces the other two forms where the denominators are nonzero.
sin^2(theta)+cos^2(theta)=1; 1+tan^2(theta)=sec^2(theta); 1+cot^2(theta)=csc^2(theta)Angle sums and differences
Sine and cosine of a sum are not found by distributing the function. The angle-addition identities combine products of sine and cosine.
These formulas derive exact values such as sin(75 degrees) from familiar angles and support double-angle and half-angle identities.
sin(a+b)=sin(a)cos(b)+cos(a)sin(b); cos(a+b)=cos(a)cos(b)-sin(a)sin(b)Worked examples
Simplify (1 - cos^2 x)/sin x
Use the fundamental Pythagorean identity.
1 - cos^2 x = sin^2 xsin^2 x / sin x = sin x
Result: sin x, for sin x != 0 in the original expression
Find sin(75 degrees)
Write 75 degrees as 45 degrees + 30 degrees.
sin75 = sin45 cos30 + cos45 sin30(sqrt2/2)(sqrt3/2)+(sqrt2/2)(1/2)
Result: sin75 = (sqrt6 + sqrt2)/4
Where this idea is used
- Simplifying trigonometric equations before solving.
- Finding exact values for compound angles.
- Rewriting integrals and wave expressions.
Common mistakes
- Writing sin(a+b)=sin(a)+sin(b).
- Canceling terms across addition instead of canceling common factors.
- Dropping domain restrictions after dividing by sine or cosine.
Questions about trigonometric identities
How is an identity different from an equation?
An identity holds throughout its shared domain; an equation may hold only for selected variable values.
Do identities depend on DEG or RAD mode?
The algebraic identities hold in either consistent angle unit, though calculus identities assume radians.
Can I verify an identity with sample values?
Samples can reveal a false claim but cannot prove it for every allowed angle; an algebraic derivation is needed.