i think it's the “dual” of $\mathrm{sinc}(x)$, that is, $$f(x) = \begin{cases} \frac 1 2, & \text{if $x=0$} \\ \frac{1-\cos{x}}{x^2}, & \text{otherwise} \end{cases}$$
they share some properties:
both have definite integral $\int_{-\infty}^\infty\frac{\sin{x}}{x}dx = \int_{-\infty}^\infty\frac{1-\cos{x}}{x^2}dx = \pi$
on taylor series, both could be regarded as chopping the constant term then divided by x to "the order the first term"-th power, $\frac 1{1!} - \frac{x^2}{3!} + \frac{x^4}{5!} - \frac{x^6}{7!} + ...$ for the sinc case, and $\frac 1{2!} - \frac{x^2}{4!} + \frac{x^4}{6!} - \frac{x^6}{8!} + ...$ for the cosine case.
it also appears in my recent (humble) study about 3D rotation. in a nutshell, to generate a rotation matrix from a rotation vector, let $$\text{rotation vector }\vec{r}=\begin{bmatrix} x\\y\\z \end{bmatrix}, \theta = \lVert \vec{r}\rVert$$ $$\mathbf{C}=\text{“cross product matrix” of }\vec{r}=\begin{bmatrix} 0&-z&y\\z&0&-x\\-y&x&0 \end{bmatrix}$$ $$\mathbf{R}=\exp(\mathbf{C}) = \mathbf{I} + \frac{\sin{\theta}}{\theta} \mathbf{C} + \frac{1-\cos{\theta}}{\theta^2}\mathbf{C}^2$$
(note that $\mathbf{C}^3=-\theta^2\mathbf{C}$)
(also $\mathbf{R}=\exp(\theta\mathbf{K}) = \mathbf{I} + (\sin{\theta})\mathbf{K} + (1-\cos{\theta})\mathbf{K}^2$ where $\mathbf{K}$ is from the normalized rotation axis, see wikipedia)
where $\frac{\sin\sqrt{x}}{\sqrt{x}}$ and $\frac{1-\cos{\sqrt{x}}}{x}$ expands similarly and adds up to the whole pretty series of $\exp{(\mathbf C)}$, allowing us to bypass the normalization step.
btw, searching for the definition only gives a bunch of tutorials for evaluating its limit when $x \to 0$ (fyi, $\frac 1 2$).
there's also a related question asking for the dual.
sorry for broken English...