reciprocal of a Fourier Cosine series

Question

Assume that $a(t)$ is an even periodic function such that $1<a(t)<2$, and is continuously differentiable everywhere. Let its Fourier series expansion be $$a(t) = a_0+a_1 cos(t)+a_2 cos(2t)+a_3 cos(3t)+...; 0<t<2\pi$$ Is there a known explicit formula for the Fourier coefficients $b_k$ of its reciprocal function $$b(t) = \frac {1}{a(t)} = b_0+b_1 cos(t)+b_2 cos(2t)+b_3 cos(3t)+...$$ in terms of the coefficients $a_k$?