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Example 1: Quadratic approximations to ln(1 + tan x) at a = 0

We can compute this by differentiating twice


Other method: Substitute and expand using known results

Set f(x) = tanx = sinx / cosx

The quadratic approximation to f at a = 0 can be obtained from those for sin x and cos x.

Since the quadratic approximation has no linear term, and after multiplying by x the quadratic term becomes cubic, we get