The following identity indicates a relationship between all trigonometric ratios. For applications to specific functions, the following infinite product formulas for trigonometric functions are useful:[42][43] With these identities, it is possible to express each trigonometric function as another (up to a plus or minus sign): The values of the trigonometric functions of these angles θ , θ ′ {displaystyle theta ,;theta ^{prime }} for some angles α {displaystyle alpha } satisfy simple identities: Either they are identical, or they have opposite signs, or they use the complementary trigonometric function. These are also called discount formulas. [2] The trigonometric identities we will examine in this section date back to a Persian astronomer who died around 950 AD. But the ancient Greeks discovered the same formulas much earlier, giving them in the form of agreements. These are special equations or postulates that apply to all values entered into the equations and have countless applications. The following table shows the trigonometric functions and their inverses to the exponential function and the complex logarithm. These identities are useful whenever expressions need to be simplified with trigonometric functions. An important application is the integration of non-trigonometric functions: a common technique is to first use the substitution rule with a trigonometric function, and then to simplify the resulting integral with a trigonometric identity. These identities are summarized in the first two rows of the following table, which also includes the sum and difference identities for the other trigonometric functions.

These formulas are useful for proving many other trigonometric identities. For example, that ei(θ+φ) = eiθ eiφ means that the angle addition formulas express trigonometric functions of angular sums as functions of and. The basic formulas for angular addition in trigonometry are given by The following identities give the result of the composition of a trigonometric function with an inverse trigonometric function. [44] The formulas for cos(A + B), sin(A − B), etc. are important but difficult to remember. Yes, you can derive them by strictly trigonometric means. But such tests are long, too difficult to reproduce when you are in the middle of an exam or a long calculation. If these values are inserted into the statement of Ptolemy`s theorem, | A C ̄ | ⋅ | B D ̄ | = | A B ̄ | ⋅ | C D ̄ | + | A D ̄ | ⋅ | B C ̄ | {displaystyle| {overline {AC}}|cdot | {overline {BD}}|=| {overline {AB}}|cdot | {overline {CD}}|+| {overline {AD}}|cdot | {overline {BC}}|} The sum of the angles is the trigonometric identity for the sine: sin ( α + β ) = sin α cos β + cos α sin β {displaystyle sin(alpha +beta )=sin alpha cos beta +cos alpha sin beta }. The angular difference formula for sin ( α − β ) {displaystyle sin(alpha -beta )} can be derived in the same way by letting the page C D ̄ {displaystyle {overline {CD}}} instead of B D ̄ {displaystyle {overline {BD}}} serve as the diameter.

[17] How to measure the height of a mountain? What about the distance between the Earth and the Sun? As with many seemingly impossible problems, we rely on mathematical formulas to find the answers. Trigonometric identities commonly used in mathematical proofs have had real-world applications for centuries, including their use in calculating large distances. By studying the unit circle, the following properties of trigonometric functions can be determined. In this section, we will learn techniques that allow us to solve problems such as those mentioned above. The following formulas simplify many trigonometric expressions and equations. Note that the term formula is used interchangeably with the word identity in this section. Finding the exact value of the sine, cosine or tangent of an angle is often easier if we can describe the given angle as two angles with known trigonometric values. We can use the special angles that we can check in the unit circle shown in (figure). There is a formula for calculating trigonometric identities for the third angle, but the zeros of the cubic equation 4×3 − 3x + d = 0 must be found, where x {displaystyle x} is the value of the cosine function in the third angle and d is the known value of the full-angle cosine function.

However, the discriminant of this equation is positive, so this equation has three real roots (only one of which is the third angle cosine solution). None of these solutions are reducible to a real algebraic expression because they use complex intermediate numbers under the cubic roots. Ptolemy`s theorem is important in the history of trigonometric identities because it proved for the first time results corresponding to molecular and differential formulas for sine and cosine (see the section on classical antiquity on the History of trigonometry page). It indicates that in a cyclic quadrilateral A B C D {displaystyle ABCD}, as shown in the figure opposite, the sum of the products of the lengths of the opposite sides is equal to the product of the lengths of the diagonals. In special cases where one of the diagonals or sides is a diameter of the circle, this theorem leads directly to the angular sum and the difference of trigonometric identities. [16] The relationship follows more easily when the circle is constructed to have a diameter of one, as shown here. In trigonometry, trigonometric identities are similarities that include trigonometric functions and apply to any value of the occurring variable for which both sides of equality are defined. Geometrically, they are identities that contain certain functions from one or more angles. They are different from triangular identities, which are identities that can include angles, but also side lengths or other lengths of a triangle. First, write the formula for the cosine of the difference of two angles. Then replace the specified values.

The first four are known as prosthapheresis formulas or sometimes Simpson`s formulas.