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The trigonometric identity gives $\sin x\cos x=\frac{1}{2}\sin 2x$, and the complementary-angle trigonometric identity gives $\cos (\pi /2-x)=\sin x$. Thus the Fourier series is already the finite expressionThe only nonzero Fourier coefficients are $b_1=1$ and $b_2=1/2$.

The binomial series beginsEquating coefficients gives $na=-2$ and $n(n-1)a^2/2=3$. Since $na\ne 0$, substitute $a=-2/n$ into the second equation to obtain $2(n-1)/n=3$, so $n=-2$ and $a=1$. ThereforeIndeed $(1+x{)}^{-2}=1-2x+3x^2-\cdots$ for $|x|<1$.

Divide by four and use completing the square:This is the standard equation of a circle, so