Cosinefourth

Cosinefourth refers to the function cos^4(x), the fourth power of the cosine of x. It is a real-valued, continuous, and infinitely differentiable function defined for all real x. The function is periodic with period π, since cos(x + π) = −cos(x) and raising to the fourth power removes the sign.

Basic properties include its evenness and its range, which lies between 0 and 1: 0 ≤ cos^4(x) ≤ 1. It attains the value 1 at x = kπ and 0 at x = π/2 + kπ, where k is an integer.

A standard expansion uses the double-angle identity. Since cos^2(x) = (1 + cos 2x)/2, squaring gives cos^4(x) = [(1 + cos 2x)/2]^2 = 3/8 + (1/2) cos 2x + (1/8) cos 4x. Equivalently, cos^4(x) = (3 + 4 cos 2x + cos 4x)/8. This form is often useful in integration and Fourier analysis.

In terms of integration, the average value of cos^4(x) over a full period is 3/8. An antiderivative is ∫ cos^4(x) dx = (3x/8) + (sin 2x)/4 + (sin 4x)/32 + C, whose derivative matches cos^4(x).

Applications of cos^4(x) appear in Fourier series, signal processing, and windowing concepts, where even-powered cosines provide simple, nonnegative, periodic components. It also serves as a basic example of how powers of trigonometric functions can be expressed as sums of cosines with multiple angles.