Identity (mathematics)

Geometrically, trigonometric identities are identities involving certain functions of one or more angles.[5] They are distinct from triangle identities, which are identities involving both angles and side lengths of a triangle. Only the former are covered in this article.

These identities are useful whenever expressions involving trigonometric functions need to be simplified. Another important application is the integration of non-trigonometric functions: a common technique which involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.

The following identities hold for all integer exponents, provided that the base is non-zero:

Unlike addition and multiplication, exponentiation is not commutative. For example, 2 + 3 = 3 + 2 = 5 and 2 · 3 = 3 · 2 = 6, but 23 = 8 whereas 32 = 9.

Also unlike addition and multiplication, exponentiation is not associative either. For example, (2 + 3) + 4 = 2 + (3 + 4) = 9 and (2 · 3) · 4 = 2 · (3 · 4) = 24, but 23 to the 4 is 84 (or 4,096) whereas 2 to the 34 is 281 (or 2,417,851,639,229,258,349,412,352). When no parentheses are written, by convention the order is top-down, not bottom-up:

Several important formulas, sometimes called logarithmic identities or log laws, relate logarithms to one another:[a]

The logarithm logb(x) can be computed from the logarithms of x and b with respect to an arbitrary base k using the following formula:

Typical scientific calculators calculate the logarithms to bases 10 and e.[6] Logarithms with respect to any base b can be determined using either of these two logarithms by the previous formula:

Given a number x and its logarithm logb(x) to an unknown base b, the base is given by:

The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities. In fact, Osborn's rule[7] states that one can convert any trigonometric identity into a hyperbolic identity by expanding it completely in terms of integer powers of sines and cosines, changing sine to sinh and cosine to cosh, and switching the sign of every term which contains a product of an even number of hyperbolic sines.[8]

The Gudermannian function gives a direct relationship between the trigonometric functions and the hyperbolic ones that does not involve complex numbers.

So, these formulas are identities in every monoid. As for any equality, the formulas without quantifier are often called equations. In other words, an identity is an equation that is true for all values of the variables.[9][10]