Using a given identity to solve for the value of an expression This problem caught my eye in the book yesterday. Till now I still get stuck. Here it is: 
If $$\frac{x}{x^2+1}=\frac{1}{3},$$ what is the value of $$\frac{x^3}{x^6+x^5+x^4+x^3+x^2+x+1}?$$
The denominator is a cyclotomic polynomial which can be expressed as $$\frac{x^7-1}{x-1}$$ but I have no idea if this even helps.

It can also solved by this equation $x^2+1=3x$.
$$\frac{x^3}{x^6+x^5+x^4+x^3+x^2+x+1}=\frac{x^3}{3x^5+x^5+x^3+3x+x}=
\frac{x^3}{4x^5+4x^3+4x-3x^3}=\frac{x^3}{12x^4+4-3x^3}= \cdots $$
The steps is natural..., I omit; however the answer given by Jack is better.
