  In this article we show a tecnique based on the Weierstrass product for the
sine and cosine function and the bisection formula for the cotangent function
that leads to a generalization of the classical Shafer-Fink inequality $
\frac{3 x}{1+2\sqrt{1+x^2}} < \arctan x < \frac{\pi x}{1+2\sqrt{1+x^2}}$. Other
algebraic approximations are also shown, including one that follows from the
Chebyshev expansion for the arctangent function in the interval $[-1,1]$.
