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.

The first identity gives:
$$ x+\frac{1}{x}=3, $$
hence:
$$ x^2+\frac{1}{x^2} = \left(x+\frac{1}{x}\right)^2 - 2 = 7, $$
$$ x^3+\frac{1}{x^3} = \left(x+\frac{1}{x}\right)^3-3\left(x+\frac{1}{x}\right)= 18 $$
and
$$ x^3+x^2+x+1+\frac{1}{x}+\frac{1}{x^2}+\frac{1}{x^3} = 18+7+3+1 = 29 $$
so the answer to your question is $\frac{1}{29}$.
