Trig Identities Calculator

Trigonometric identities are equations involving trigonometric functions that are true for all valid values of the variable. Common examples include the Pythagorean identity sin²(x) + cos²(x) = 1, double-angle identities like sin(2x) = 2sin(x)cos(x), and sum/difference identities such as sin(A+B) = sin(A)cos(B) + cos(A)sin(B).

The Pythagorean identity states that sin²(θ) + cos²(θ) = 1 for any angle θ. This comes directly from the Pythagorean theorem applied to a unit circle, where the x-coordinate is cos(θ) and the y-coordinate is sin(θ), and the radius is always 1.

Select 'Degrees' if your angle is measured in the familiar 0–360° scale, or 'Radians' if it's expressed in terms of π (e.g., π/6 ≈ 0.5236 for 30°). The calculator automatically converts your input before computing all identities.

tan(θ) = sin(θ)/cos(θ) and becomes undefined when cos(θ) = 0, which occurs at 90°, 270°, etc. Similarly, cot(θ) is undefined when sin(θ) = 0. Very large displayed values indicate the function is approaching infinity near these angles.

Double-angle identities express trig functions of 2A in terms of functions of A. For example, sin(2A) = 2sin(A)cos(A) and cos(2A) = cos²(A) − sin²(A). They are widely used in calculus integrations, solving trig equations, and simplifying expressions in physics and engineering.

Sum and difference identities allow you to compute trig functions of the sum or difference of two angles. For instance, sin(A+B) = sin(A)cos(B) + cos(A)sin(B). Enter both Angle A and Angle B in this calculator to see these results computed automatically.

These are the three reciprocal trigonometric functions: sec(θ) = 1/cos(θ), csc(θ) = 1/sin(θ), and cot(θ) = 1/tan(θ) = cos(θ)/sin(θ). They arise frequently in calculus, integration by substitution, and advanced geometry problems.

Yes — plug in specific angle values and compare the left-hand and right-hand sides of your identity using the displayed outputs. If both sides produce the same numeric result across multiple angle values, the identity is likely valid. For a formal algebraic proof, you would need to manipulate the expression symbolically.