Cos(2pi) is equal to 1. The cosine function, cos(x), oscillates between 1 and -1 with a period of 2pi as x varies. By definition, cos(0) = 1, and the periodicity of the function means the cosine of all multiples of 2pi (2pi, 4pi and so on) is also equal to 1.
By definition, cos(x) is equal to the ratio of the adjacent side of a right-angled triangle to its hypotenuse. As the angle of the triangle approaches zero, this ratio tends towards one. An angle of zero is always equivalent to an angle of 2pi radians, so cos(2pi) = cos(0) = 1.