cos2n

Cos2n denotes the cosine of twice the angle n, written as cos(2n). In standard mathematics, angles are measured in radians, and the cosine function is periodic with period 2π. Consequently, as a function of a real variable n, cos(2n) has period π. When n is restricted to integers, however, the sequence n ↦ cos(2n) has no finite period because the ratio 2/π is irrational; the values are instead dense in the interval [-1, 1].

A key property is the double-angle identity: cos(2n) = cos^2 n − sin^2 n, which can also be written as cos(2n) = 2 cos^2 n − 1 and cos(2n) = 1 − 2 sin^2 n. These relations connect cos(2n) to cos n and sin n and hold for all real n. In terms of Chebyshev polynomials, cos(2n) = T_2(cos n), where T_2(x) = 2x^2 − 1.

Cos(2n) also admits a complex exponential representation: cos(2n) = (e^{i2n} + e^{−i2n})/2, which is often useful in Fourier analysis and signal processing. For numerical illustration, cos(2n) yields values such as cos(0) = 1, cos(2) ≈ −0.4161, cos(4) ≈ −0.6536, and cos(6) ≈ 0.9605.

See also: Trigonometric functions, double-angle identities, Chebyshev polynomials, and discrete cosine sequences.