Trigonometric Identity Calculator
Verify an identity or simplify one side while seeing the exact rule behind every rewrite.
Drag & drop or upload an image or PDF
Enter a full identity to verify, or start with “simplify” followed by one trigonometric expression.
Try an example
An identity must work across its common domain
A trigonometric identity is an equality that holds for every angle where both sides are defined. That is different from a trigonometric equation, which may be true only at particular angles. This calculator focuses on proving identities and simplifying expressions, not on finding selected angle solutions.
A clean proof usually starts with the side that has more structure. Algebra, reciprocal and quotient identities, and the Pythagorean identities often reduce it to the other side. Each rewrite should remain equivalent on the original domain, so canceled factors and undefined trig functions still matter.
How to use the trigonometric identity calculator
Enter the problem as written
Type or paste the full expression. You can also upload a clear photo or PDF and check the extracted text before solving.
Read the working, not only the answer
Each transformation is separated and explained so you can compare it with your own method.
Ask about any step
Continue in the same solution to request another method, check a restriction, or ask why a rule applies.
Core Pythagorean identities
These three identities connect the six trigonometric functions and are often the shortest route through a simplification.
A reliable way to work through it
Work on the more complicated side
Rewrite one side until it matches the other. Changing both sides at once can hide a circular argument.
Convert strategically
Use reciprocal, quotient, and Pythagorean identities before converting every function to sine and cosine.
Track the common domain
Record values excluded by tangent, secant, cotangent, cosecant, denominators, or canceled factors.
Worked example
Simplify (1 - cos²x) / sin x
Rearrange the Pythagorean identity sin²x + cos²x = 1.
Replace the numerator with an equivalent expression.
Cancel one sine factor while keeping the original denominator restriction.
The original expression is undefined wherever sin x equals zero.
On the domain of the original expression, the simplified form is sin x. The excluded values are x = kπ for every integer k.
Common mistakes to check
Testing a few angles as proof
Numerical checks can catch a false claim, but only equivalent symbolic rewrites prove an identity.
Changing both sides together
A valid proof transforms one side into the other without assuming the equality it is trying to establish.
Canceling through addition
Factors can cancel from a product or quotient. Terms separated by addition or subtraction cannot.
Dropping excluded values
A simpler final expression may be defined at angles where the original expression was not.
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Questions students ask
What is the difference between an identity and an equation?
An identity is true for every value in the common domain. An equation can be true only for particular values of the variable.
Can numerical substitution prove a trig identity?
No. It is useful as a check or counterexample search, but a proof requires algebraically equivalent steps.
Why does the answer include domain restrictions?
Canceling a factor can make the final form look defined at points where the original denominator was zero. Those exclusions remain part of the result.
Which side of an identity should I simplify first?
Start with the side containing more fractions, functions, or operations. It is usually easier to reduce structure than to build it.