Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2005 / Paper 1 / 1 / a / Solution

The Taylor series $\sin x=x-x^3/6+O(x^5)$ gives ${\sin }^{2}x=x^2(1-x^2/3+O(x^4))$. Taking its reciprocal and cancelling the singular term givesThus the apparent singularity at zero is removable, and the first nonzero term is the constant $1/3$.