Proofs of trigonometric identities

There are several equivalent ways for defining trigonometric functions, and the proof of the trigonometric identities between them depend on the chosen definition. The oldest and somehow the most elementary definition is based on the geometry of right triangles. The proofs given in this article use this definition, and thus apply to non-negative angles not greater than a right angle. For greater and negative angles, see Trigonometric functions. Other definitions, and therefore other proofs are based on the Taylor series of sine and cosine, or on the differential equation to which they are solutions. The six trigonometric functions are defined for every real number, except, for some of them, for angles that differ from 0 by a multiple of the right angle (90°). Referring to the diagram at the right, the six trigonometric functions of θ are, for angles smaller than the right angle: In the case of angles smaller than a right angle, the following identities are direct consequences of above definitions through the division identity They remain valid for angles greater than 90° and for negative angles. Or Two angles whose sum is π/2 radians (90 degrees) are complementary. In the diagram, the angles at vertices A and B are complementary, so we can exchange a and b, and change θ to π/2 − θ, obtaining: Pythagorean trigonometric identity Identity 1: The following two results follow from this and the ratio identities. To obtain the first, divide both sides of by ; for the second, divide by . Similarly Identity 2: The following accounts for all three reciprocal functions. Proof 2: Refer to the triangle diagram above. Note that by Pythagorean theorem. Substituting with appropriate functions - Rearranging gives: List of trigonometric identities#Angle sum and difference identities Draw a horizontal line (the x-axis); mark an origin O. Draw a line from O at an angle above the horizontal line and a second line at an angle above that; the angle between the second line and the x-axis is . Place P on the line defined by at a unit distance from the origin.