Cofunction Identities
The complete cofunction identity list with explanations, complementary angle logic, and practical simplification strategies.
By Cofunction Calculator Team May 23, 2026
Quick Answer
- Short definition
- Cofunction identities are equations that equate a trigonometric function at angle θ with its partner function at 90° - θ.
- Formula
- csc(90° - θ) = sec(θ)
Introduction
Cofunction Calculator applies all six identity pairs automatically from your input, which makes this identity set easy to test while you study.
Cofunction identities are core reference tools in trigonometry. You will use them in simplification, proof steps, and unit circle reasoning throughout algebra and precalculus courses.
Main Content
What is it?
A cofunction identity is a guaranteed equivalence between paired trigonometric functions across complementary angles. These identities are not approximations. They hold exactly wherever both sides are defined.
If you are new to the concept, start with What Is a Cofunction? and then use this article as your complete identity reference sheet.
Formula
Full identity list:
- sin(x) = cos(90° - x)
- cos(x) = sin(90° - x)
- tan(x) = cot(90° - x)
- cot(x) = tan(90° - x)
- sec(x) = csc(90° - x)
- csc(x) = sec(90° - x)
For geometric interpretation of these identities, continue with Cofunctions in Trigonometry.
Step-by-step guide
- Locate the function in your expression.
- Match it to its cofunction partner.
- Replace x with 90° - x in the partner form.
- Simplify the resulting expression.
- Confirm using known values or calculator checks.
Example
Simplify sec(20°) using cofunction identity.
Since sec(x) = csc(90° - x), we get sec(20°) = csc(70°). This form is often easier when nearby terms involve cosecant.
Verify both sides numerically to confirm the transformation.
FAQ
They hold algebraically where both sides are defined.
No. Cofunction identities use complementary angle partners, while reciprocal identities use multiplicative inverses.
Conclusion
Keep this identity list accessible during practice sessions. Most simplification speed comes from recognizing which pair applies in one glance.
Test each identity pair in the calculator to reinforce memory with numeric confirmation.