cosxsinx

Cosxsinx denotes the product of the cosine and sine of the same real variable x, written as f(x) = cos x · sin x. Using the double-angle identity sin 2x = 2 sin x cos x, this function can be expressed as f(x) = (1/2) sin 2x, highlighting its periodicity and range.

The function has period π, since sin 2x has period π. Its range is [-1/2, 1/2]. Zeros occur when sin 2x = 0, i.e., at x = nπ/2 for integers n. Maximum values of 1/2 occur at x = π/4 + nπ, while minimum values of -1/2 occur at x = 3π/4 + nπ. The graph is an odd function, with f(-x) = -f(x).

Derivatives and integrals follow standard rules. f'(x) = cos^2 x − sin^2 x = cos 2x. An antiderivative is ∫ cos x sin x dx = (1/2) sin^2 x + C (or equivalently −(1/2) cos^2 x + C).

Beyond these relations, cos x sin x often appears in product-to-sum transformations and trigonometric simplifications, serving as a compact way to represent sine and cosine products. It also provides a simple example of how double-angle functions govern the behavior of product expressions, with straightforward geometric and algebraic interpretations.