cosxcosx

Cosxcosx denotes the product cos x cos x, i.e., cos^2 x. It is a basic trigonometric expression that arises in algebra, calculus, and physics, and it is often encountered in identities and integrals.

One of the simplest identities is cos^2 x = (1 + cos 2x)/2, obtained from the double-angle formulas. It is also equal to 1 − sin^2 x, by the Pythagorean identity sin^2 x + cos^2 x = 1.

As a function of x, cos^2 x takes values between 0 and 1 for all real x, and it has a period of π because cos^2(x + π) = cos^2 x. The graph repeats every π and remains nonnegative.

The derivative is d/dx cos^2 x = −sin 2x, since the derivative of cos x is −sin x and by the chain rule. An antiderivative is ∫ cos^2 x dx = x/2 + sin 2x/4 + C, which follows from the same double-angle identity.

In applications, cos^2 x appears in Fourier analysis, wave and signal processing, and in various trigonometric integrals. It also provides a convenient form in problems that involve averaging or energy estimates over a period, thanks to its relation to cos 2x and its bounded range.