Show that $C\frac{dC}{dr}\ + S\frac{dS}{dr}\ = (C^2 + S^2)\cos{\theta}$ Given $$C=1+r\cos{\theta}\ +\frac{r^2\cos{2\theta}}{2!}\ + \frac{r^3\cos{3\theta}}{3!}\ + \dotsb$$ and $$S = r\sin{\theta}\ + \frac{r^2\sin{2\theta}}{2!}\ + \frac{r^3\sin{3\theta}}{3!}\ + \dotsb$$
Show the following$$C\frac{dC}{dr}\ + S\frac{dS}{dr}\ = (C^2 + S^2)\cos{\theta}$$
I am currently solving the problems given in Differential Calculus for Beginners by Joseph Edwards. As a beginner I am completely clueless about the approach to the above question. I tried to find the answer keys to this book, but sadly non exist on the internet.

$$
C\frac{dC}{dr}+S\frac{dS}{dr}=C\sum_{n=1}^\infty\frac{nr^{n-1}\cos n\theta}{n!}+S\sum_{n=1}^\infty\frac{nr^{n-1}\sin n\theta}{n!}\\
=C\sum_{n=1}^\infty\frac{ r^{n-1}\cos n\theta}{(n-1)!}+S\sum_{n=1}^\infty\frac{ r^{n-1}\sin n\theta}{(n-1)!}=C\sum_{n=0}^\infty\frac{ r^{n }\cos (n+1)\theta}{ n !}+S\sum_{n=0}^\infty\frac{ r^{n }\sin (n+1)\theta}{ n !}\\
=C\sum_{n=0}^\infty\frac{ r^ n }{ n !}(\cos n\theta\cos\theta-\sin n\theta\sin\theta )+S\sum_{n=0}^\infty\frac{ r^ n }{ n !}(\sin n\theta\cos\theta+\cos n\theta\sin\theta )\\
= C^2 \cos\theta-C S\sin\theta+SC\sin\theta+S^2 \cos\theta
=(C^2+S^2)\cos\theta
$$
