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The function powcos(p) returns the formal power series that is equivalent to cos(p).

The powseries function evalpow accepts the form in either powcos, Cos, or cos.

The command with(powseries,powcos) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{powseries}\right)\:$

$a≔\mathrm{powcos}\left(x+1\right)\:$

$b≔\mathrm{tpsform}\left(a\,x\,5\right)$

$b≔\cos \left(1\right)-\sin \left(1\right)x-\frac{1}{2}\cos \left(1\right)x^2+\frac{1}{6}\sin \left(1\right)x^3+\frac{1}{24}\cos \left(1\right)x^4+O\left(x^5\right)$

$c≔\mathrm{evalpow}\left(\mathrm{Cos}\left(\log \left(1+x\right)\right)\right)\:$

$d≔\mathrm{tpsform}\left(c\,x\,5\right)$

$d≔1-\frac{1}{2}{x}^2+\frac{1}{2}{x}^3-\frac{5}{12}{x}^4+O\left(x^5\right)$

$e≔\mathrm{powdiff}\left(\mathrm{powcos}\left(x\right)\right)\:$

$f≔\mathrm{tpsform}\left(e\,x\,6\right)$

$f≔-x+\frac{1}{6}{x}^3-\frac{1}{120}{x}^5+O\left(x^6\right)$

$g≔\mathrm{evalpow}\left(\tan \left(x^2\right)\right)\:$

$h≔\mathrm{tpsform}\left(g\,x\,10\right)$

$h≔x^2+\frac{1}{3}{x}^6+O\left(x^{10}\right)$