Every trig identity you need in one place, organized by category, with an interactive verifier that tests expressions numerically.
Every trigonometric identity you'll ever need — organized by category. Plus an interactive verifier that tests expressions numerically across multiple angles to confirm they're equivalent.
Derived from x² + y² = 1 on the unit circle
Each trig function has an inverse partner
Express tangent and cotangent in terms of sine and cosine
Express trig functions of 2θ in terms of θ
Express trig functions of θ/2 in terms of θ
Combine or decompose angles (using x and 1 as angle variables)
Convert products of trig functions into sums
Complementary angle relationships (π/2 = 90°)
Start with just one: sin²θ + cos²θ = 1. Divide both sides by cos²θ and you get the tangent-secant identity. Divide by sin²θ and you get the cotangent-cosecant identity. Three identities from one!
Trigonometric identities are equations involving trigonometric functions that are true for every valid input value. Unlike equations that are true for only specific values of the variable (like 2x = 6 being true only when x = 3), an identity like sin²(θ) + cos²(θ) = 1 holds for every angle θ. Identities are powerful tools for simplifying expressions, solving equations, and proving mathematical results.
These identities are not arbitrary rules to memorize. They arise naturally from the geometry of the unit circle and the definitions of the trig functions. Once you understand the underlying connections, many identities can be derived from just a few foundational ones.
The most important trigonometric identity is the Pythagorean identity, which comes directly from the equation of the unit circle: x² + y² = 1. Since x = cos(θ) and y = sin(θ) on the unit circle, we get:
By dividing both sides by cos²(θ), you obtain 1 + tan²(θ) = sec²(θ). Dividing by sin²(θ) instead yields 1 + cot²(θ) = csc²(θ). This demonstrates how three seemingly separate identities are really one identity in disguise.
The double angle formulas express trig functions of 2θ in terms of θ. They are derived from the sum formulas by setting both angles equal. The most commonly used are:
The cosine double angle formula has two alternate forms: cos(2θ) = 2cos²(θ) - 1 and cos(2θ) = 1 - 2sin²(θ). These alternate forms are obtained by substituting the Pythagorean identity, and they give rise to the half-angle formulas when solved for sin(θ/2) and cos(θ/2).
Simplify the expression (1 - cos(2x)) / sin(2x):
The entire expression simplifies to tan(x). You can verify this using the interactive verifier above by entering the left and right sides.
The verifier tests whether two expressions are equivalent by evaluating both sides at 10 different angles and checking whether the results match to within a small tolerance. Enter any expression using the supported syntax: sin(x), cos(x), tan(x), csc(x), sec(x), cot(x), with standard arithmetic operators. Use ^ for powers and pi for the constant π. Click "Test It" on any identity in the reference sheet to load it directly into the verifier.