Choose an identity mode
Select a tool, load an example, or enter your own values.
Understanding identities
Why identities matter
Trigonometric identities are equations that stay true for every angle where both sides are defined. They let you transform expressions, verify equations, solve trig problems, and simplify advanced calculus work. A good identity calculator should show both the formula and the substitution steps.
Pythagorean identities
The identity sin²θ + cos²θ = 1 comes from the unit circle. The terminal point has coordinates (cosθ, sinθ). Since every unit-circle point is one unit from the origin, the coordinate equation becomes cos²θ + sin²θ = 1.
Angle identities
Sum, difference, double-angle, and half-angle identities rewrite one angle expression in another form. These are helpful when exact angles are built from familiar angles, such as 75° = 45° + 30° or 15° = 45° − 30°.
Product and sum transformations
Product-to-sum and sum-to-product identities change the structure of a trig expression. They are common in integration, wave analysis, signal processing, and equation solving. The calculator compares both sides numerically so the transformation is clear.
Verification approach
Numeric verification is not a formal symbolic proof, but it is excellent for checking substitutions and learning the pattern. The step cards show the identity used, the values substituted, and the difference between both sides.
FAQs
What does this identities calculator do?
It applies and verifies common trigonometric identities with substitutions, numeric checks, formula cards, visual previews, and downloadable result tables.
Can it prove identities symbolically?
It gives formula-based transformations and numeric verification. It does not perform unrestricted symbolic algebra on any typed expression.
Which identities are included?
It includes Pythagorean, reciprocal, quotient, cofunction, even/odd, sum, difference, double-angle, half-angle, product-to-sum, sum-to-product, and power-reducing identities.
Can I use radians?
Yes. Select radians and enter decimals, fractions, pi, pi/6, or similar values.
Why is tangent undefined sometimes?
Tangent equals sine divided by cosine. It is undefined when cosine is zero.
Why does half-angle need a branch sign?
Half-angle formulas use a square root. The sign depends on the quadrant of the half angle.
What is numeric verification?
Numeric verification substitutes angle values into both sides and checks whether the results match within rounding tolerance.
Are reciprocal identities different from quotient identities?
Yes. Reciprocal identities use inverse pairs like sin and csc. Quotient identities express tan and cot using sine and cosine.
What are cofunction identities?
They relate functions of complementary angles, such as sinθ = cos(90° − θ).
When should I use product-to-sum?
Use it when a product of trig functions needs to be converted into sums or differences. It is common in integration.
When should I use sum-to-product?
Use it when added trig terms need to be factored into a product form.
Can I download results?
Yes. Use the CSV button for the result table or the print button to save as a PDF.