cos2x2

Cos2x2 is not a standard mathematical term. In many contexts it may be read as the square of the cosine of 2x, written as cos^2(2x) or (cos(2x))^2. Other possible readings include cos(2x^2) or a typographical miswrite, so precise notation with parentheses is recommended to avoid ambiguity when discussing this expression.

If the intended meaning is cos^2(2x), then it can be simplified using the double-angle identity for cosine. Since cos(4x) = 2 cos^2(2x) − 1, one obtains cos^2(2x) = (1 + cos(4x))/2. This “power-reduction” form is often convenient in integration, Fourier analysis, or when combining with other trigonometric terms. Related identities include sin^2(2x) = (1 − cos(4x))/2 and the fundamental Pythagorean relation cos^2(2x) + sin^2(2x) = 1.

The function cos^2(2x) takes values between 0 and 1 for all real x, since cos(2x) lies in [−1, 1] and squaring yields a range [0, 1]. Its graph is nonnegative and has a period of π, reflecting the period of cos(2x) halved by the squaring operation. It exhibits peaks at x where cos(2x) = ±1 and zeros where cos(2x) = 0.

In practice, to avoid ambiguity, it is recommended to write cos^2(2x) or (cos(2x))^2 rather than a compact string like cos2x2. The expression frequently appears in trigonometric integrals, signal processing, and problems involving power reductions or trigonometric expansions.