Are there any elementary functions $\beta(x)$ that follows this integral $\int_{y-1}^{y} \beta(x) dx =\cos(y)$ Are there any simple functions $\beta(x)$ that follows this integral $$\int_{y-1}^{y} \beta(x) dx =\cos(y)$$
I think there is an infinite amount of solutions that are continuous everywhere but how can I find one that only uses elementary functions?

Try this:
$$
\beta(x) = \frac{\sin(x+1)}{2} - \frac{\sin(1)\;\cos(x+1)}{2\cos(1)-2} 
$$

How to find this?  If $N$ is a positive integer, then
$$
\int_0^N \beta(x)\;dx = \sum_{n=1}^N\int_{n-1}^n\beta(x)\;dx
=\sum_{n=1}^N \cos n
=-\frac{\cos(N+1)}{2}-\frac{\sin(2)\;\sin(N+1)}{2(\cos(1)-1)} - \frac{1}{2}
$$
Now make a guess that the same formula will also work when we
replace $N$ with $y$ which is not
a positive integer.  Differentiate to get $\beta(y)$.
Then check that it works.

Check:
\begin{align}
\beta(x) &= \frac{\sin(x+1)}{2} - \frac{\sin(1)\;\cos(x+1)}{2\cos(1)-2}
\\
\int\beta(x)\;dx &=
\frac{-\cos(x+1)}{2} - \frac{\sin(1)\;\sin(x+1)}{2\cos(1)-2}
\\
\int_{y-1}^y\beta(x)\;dx &=
\frac{-\cos(y+1)+\cos(y)}{2} 
+ \frac{\sin(1)\;(-\sin(y+1)+\sin(y))}{2\cos(1)-2}
\\ &=
\frac{-\cos(1)\cos(y)+\sin(1)\sin(y)+\cos(y)}{2} 
+ \frac{\sin(1)\;(-\cos(1)\sin(y)-\sin(1)\cos(y)+\sin(y))}{2\cos(1)-2}
\\ &=
\left[\frac{-\cos(1)+1}{2}
+\frac{-\sin^2(1)}{2(\cos(1)-1)}\right]\cos(y)
+\left[\frac{\sin(1)}{2}
+\frac{-\cos(1)\sin(1)-\sin(1)}{2(\cos(1)-1)}\right]\sin(y)
\\ &=
\frac{-\cos^2(1)+2\cos(1)-1-\sin^2(1)}{2(\cos(1)-1)}\cos(y)
+\frac{\cos(1)\sin(1)-\sin(1)-\sin(1)\cos(1)+\sin(1)}{2(\cos(1)-1)}\sin(y)
\\ &=
\frac{2\cos(1)-2}{2(\cos(1)-1)}\cos(y)
+\frac{\cos(1)\sin(1)-\sin(1)-\sin(1)\cos(1)+\sin(1)}{2(\cos(1)-1)}\sin(y)
\\ &=
\cos(y)
\end{align}
