I'm kind of puzzled by the equality $$a(z) \cos (z) + b(z) \sin (z)=c(z) \cos (z) + d(z) \sin (z)$$ If $a$, $b$, $c$, $d$ are constant, then $a=c$ and $b=d$, but is there a more general form of functions $a(z)$, ..., $d(z)$ that will satisfy this?
By the way, is there a name for these coefficient-functions?
Thanks in advance!
Consider a vector $\bf u$ and a vector $\bf n$ normal to it.
The two vectors $\bf v = \alpha \bf u + \beta \bf n$ and $\bf w = \alpha \bf u + \gamma \bf n$ have the same dot product with $\bf u$.
Taking therefore the rotating vector $\bf u(z) =(\cos z , \sin z)$ and $\bf n =(- \sin z, \cos z)$ normal to it, the two vectors $$ \eqalign{ & {\bf v} = \alpha (z){\bf u} + \beta (z){\bf n} = \left( {\alpha (z)\cos z - \beta (z)\sin z,\;\alpha (z)\sin z + \beta (z)\cos z} \right) \cr & {\bf w} = \alpha (z){\bf u} + \gamma (z){\bf n} = \left( {\alpha (z)\cos z - \gamma (z)\sin z,\;\alpha (z)\sin z + \gamma (z)\cos z} \right) \cr} $$ have the same dot product with $\bf u(z)$, i.e. their components, depending on any three functions $\alpha (z), \beta (z) \gamma (z)$, are the four coefficients $a(z),b(z),c(z),d(z)$ that you requested.