One can sometimes find a maclaurin series by the method of equating coefficients. For Example, let
tan x = sin x / cos x = a0 + a1x + a2x2 + ...
x - x3 / 6 + ... = (a0 + a1x +a2x2 + ...) (1 - x2/2 + ...)
= a0 + a1x + (a2 - a0/2)x2 + (a3 - a1/2)x3 + ...
Thus,
a0 = 0, a1 = 1, a2 - a0 / 2 = 0, a3 - a1 / 2 = - 1/6, ...
So,
a0 = 0, a1 = 1, a2 = 0, a3 = 1/3, ... and therefore,
tan x = 0 + x + 0 + 1/3 x3 + ...
Which agrees with Problem 1. Use this method to find the terms through x4 in the series for sec x?