The curious identity

is a special case of an identity that contains one variable:

A similar-looking identity is

and in addition

The following is perhaps not as readily generalized to an identity containing variables:

Degree measure ceases to be more felicitous than radian measure when we consider this identity with 21 in the denominators:

The factors 1, 2, 4, 5, 8, 10 may start to make the pattern clear: they are those integers less than 21/2 that are relatively prime to (or have no prime factors in common with) 21. The last several examples are corollaries of a basic fact about the irreducible cyclotomic polynomials: the cosines are the real parts of the zeroes of those polynomials; the sum of the zeroes is the Möbius function evaluated at (in the very last case above) 21; only half of the zeroes are present above. The two identities preceding this last one arise in the same fashion with 21 replaced by 10 and 15, respectively.

An efficient way to compute π is based on the following identity without variables, due to Machin:

or, alternatively, by using Euler's formula:

With the golden ratio φ:

Also see exact trigonometric constants.

Identities without variables
Relation to the complex exponential function
अर्ध कोणों के सूत्रों का परिवर्धन (Extension of half-angle formulae)
"cis"