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Odd/Even Identities

Odd/Even Identities
Plus/Minus Identities

Trig identities which show whether each trig function is an odd function or an even function.

 

Odd/Even Identities

sin (–x) = –sin x

cos (–x) = cos x

tan (–x) = –tan x

csc (–x) = –csc x

sec (–x) = sec x

cot (–x) = –cot x

Key Formula

sin⁡(−x)=−sin⁡xcsc⁡(−x)=−csc⁡xcos⁡(−x)=cos⁡xsec⁡(−x)=sec⁡xtan⁡(−x)=−tan⁡xcot⁡(−x)=−cot⁡x\begin{aligned} \sin(-x) &= -\sin x & \csc(-x) &= -\csc x \\ \cos(-x) &= \cos x & \sec(-x) &= \sec x \\ \tan(-x) &= -\tan x & \cot(-x) &= -\cot x \end{aligned}
Where:
  • xx = Any angle, measured in radians or degrees
  • −x-x = The negation (opposite) of the angle x

Worked Example

Problem: Use odd/even identities to find the exact value of sin(−30°) and cos(−60°).
Step 1: Apply the odd identity for sine: sin(−x) = −sin x.
sin⁡(−30°)=−sin⁡(30°)\sin(-30°) = -\sin(30°)
Step 2: Evaluate sin(30°), which is a known value.
−sin⁡(30°)=−12-\sin(30°) = -\frac{1}{2}
Step 3: Apply the even identity for cosine: cos(−x) = cos x.
cos⁡(−60°)=cos⁡(60°)\cos(-60°) = \cos(60°)
Step 4: Evaluate cos(60°), which is a known value.
cos⁡(60°)=12\cos(60°) = \frac{1}{2}
Answer: sin(−30°) = −1/2 and cos(−60°) = 1/2. The negative sign passes through sine (odd) but disappears for cosine (even).

Another Example

This example shows how to apply odd/even identities to simplify an algebraic expression involving multiple trig functions, rather than evaluating a single numerical value.

Problem: Simplify the expression sin(−x) · cos(−x) + tan(−x) into an expression with no negative angles.
Step 1: Replace sin(−x) using the odd identity.
sin⁡(−x)=−sin⁡x\sin(-x) = -\sin x
Step 2: Replace cos(−x) using the even identity.
cos⁡(−x)=cos⁡x\cos(-x) = \cos x
Step 3: Replace tan(−x) using the odd identity.
tan⁡(−x)=−tan⁡x\tan(-x) = -\tan x
Step 4: Substitute all three results into the original expression.
(−sin⁡x)(cos⁡x)+(−tan⁡x)=−sin⁡xcos⁡x−tan⁡x(-\sin x)(\cos x) + (-\tan x) = -\sin x \cos x - \tan x
Answer: The simplified expression is −sin x cos x − tan x.

Frequently Asked Questions

How do you remember which trig functions are odd and which are even?
Only cosine and secant are even — notice they are related (secant is 1/cos). All the other four functions (sine, tangent, cosecant, cotangent) are odd. A quick mnemonic: cosine starts with 'c' for 'co-even,' and its reciprocal secant shares that property. Everything else is odd.
Why is cosine an even function and sine an odd function?
On the unit circle, reflecting an angle x to −x means reflecting the point across the horizontal axis. The x-coordinate (cosine) stays the same, so cos(−x) = cos x. The y-coordinate (sine) flips sign, so sin(−x) = −sin x. This geometric reasoning is the foundation of the odd/even identities.
When do you use odd/even identities in trigonometry?
You use them whenever you encounter a negative angle inside a trig function. They are essential for simplifying expressions, verifying other identities, and evaluating trig values for negative angles. They also appear frequently when working with Fourier series and analyzing function symmetry.

Odd Trig Functions vs. Even Trig Functions

Odd Trig FunctionsEven Trig Functions
Definitionf(−x) = −f(x): the function's sign flipsf(−x) = f(x): the function's value is unchanged
Which trig functionssin, tan, csc, cotcos, sec
Graph symmetrySymmetric about the origin (rotational symmetry)Symmetric about the y-axis (mirror symmetry)
Examplesin(−45°) = −sin(45°) = −√2/2cos(−45°) = cos(45°) = √2/2

Why It Matters

Odd/even identities appear throughout precalculus and calculus, especially when simplifying integrals of trig functions over symmetric intervals. In calculus, knowing that sine is odd lets you immediately conclude that the integral of sin x from −a to a equals zero. These identities also serve as a key step in verifying more complex trig identities and solving trig equations that involve negative angles.

Common Mistakes

Mistake: Assuming all trig functions are odd because most of them are.
Correction: Cosine and secant are even, not odd. For these two, the negative sign disappears: cos(−x) = cos x and sec(−x) = sec x. Always check whether the specific function is odd or even before applying the identity.
Mistake: Confusing a negative angle with a negative output — writing cos(−x) = −cos x.
Correction: A negative input does not automatically produce a negative output. For even functions like cosine, the output is unchanged. Only for odd functions does the negative pass through to the output.

Related Terms