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
A compact identity map
An identity is true wherever both sides are defined. Domain restrictions remain important after simplification.
| Family | Identity | Useful for |
|---|---|---|
| Quotient | tan(x) = sin(x)/cos(x) | Rewriting in sine and cosine |
| Reciprocal | sec(x) = 1/cos(x) | Clearing reciprocal functions |
| Pythagorean | sin^2(x) + cos^2(x) = 1 | Replacing a squared pair |
| Angle sum | sin(a+b) = sin(a)cos(b) + cos(a)sin(b) | Exact values and expansions |
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 identity is the unit-circle equation
A point at angle x on the unit circle is (cos x, sin x). Every point on that circle satisfies X^2 + Y^2 = 1.
X = cos(x), Y = sin(x)Use the coordinate definitions.X^2 + Y^2 = 1The circle has radius 1.cos^2(x) + sin^2(x) = 1Substitute the coordinates.
Worked example 1
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
Verify before using an expression as an identity
Testing a few values cannot prove an identity, but one counterexample can disprove a false claim.
1. Is sin(2x) = 2sin(x) an identity?
Hint: Test x = 30 degrees.
Answer: No. sin(60 degrees) = sqrt(3)/2, while 2sin(30 degrees) = 1.
2. Rewrite 1 - sin^2(x).
Hint: Rearrange the Pythagorean identity.
Answer: 1 - sin^2(x) = cos^2(x).
Identity checks before canceling
- 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.