For example, consider the tangent identity, We can interpret the tangent of a negative angle as Tangent is therefore an odd function, which means that for all in the domain of the tangent function. The other even-odd identities follow from the even and odd nature of the sine and cosine functions. The identity is found by rewriting the left side of the equation in terms of sine and cosine.ДЇor all in the domain of the sine and cosine functions, respectively, we can state the following: The second and third identities can be obtained by manipulating the first. We have already seen and used the first of these identifies, but now we will also use additional identities. We will begin with the Pythagorean identities (see (Figure)), which are equations involving trigonometric functions based on the properties of a right triangle. In this first section, we will work with the fundamental identities: the Pythagorean identities, the even-odd identities, the reciprocal identities, and the quotient identities. Sometimes we have to factor expressions, expand expressions, find common denominators, or use other algebraic strategies to obtain the desired result. To verify the trigonometric identities, we usually start with the more complicated side of the equation and essentially rewrite the expression until it has been transformed into the same expression as the other side of the equation. Consequently, any trigonometric identity can be written in many ways. We already know that all of the trigonometric functions are related because they all are defined in terms of the unit circle. Basic properties and formulas of algebra, such as the difference of squares formula and the perfect squares formula, will simplify the work involved with trigonometric expressions and equations. In fact, we use algebraic techniques constantly to simplify trigonometric expressions. They are the basic tools of trigonometry used in solving trigonometric equations, just as factoring, finding common denominators, and using special formulas are the basic tools of solving algebraic equations. Identities enable us to simplify complicated expressions. Verifying the Fundamental Trigonometric Identities

Trigonometric identities