Answer
Write $y'$ in terms of $R, y, z$.
$$y'=y\cos R-z\sin R$$
Work Step by Step
$$y'=r\cos(\theta+R)\hspace{1cm}y=r\cos\theta\hspace{1cm}z=r\sin\theta$$
We now would analyze $y'$ formula.
For $\cos(\theta+R)$, the cosine sum identity can be applied:
$$\cos(\theta+R)=\cos\theta\cos R-\sin\theta\sin R$$
Therefore, $$y'=r(\cos\theta\cos R-\sin\theta\sin R)$$
$$y'=r\cos\theta\cos R-r\sin\theta\sin R$$
$$y'=(r\cos\theta)\cos R-(r\sin\theta)\sin R$$
Now recall that $y=r\cos\theta$ and $z=r\sin\theta$.
$$y'=y\cos R-z\sin R$$
That is the formula of $y'$ in terms of $R, y, z$.