
tan reduction formula
So the proof for the reduction formula for [tan(x)]^n I found here, and it makes sense
But I did this and it is very close, but not the same formula, what's wrong?
$\displaystyle \int tan^nx dx = \int tan^n^^2x *tan^2x dx $
$\displaystyle =\int tan^n^^2x(sec^2x 1) dx $
$\displaystyle = \int tan^n^^2x*sec^2x dx  \int tan^n^^2x dx$
here I did an integration by parts of the first integral above:
$\displaystyle u= tan^n^^2x,,,,,,,,,,,,,,,dv=sec^2xdx$
$\displaystyle du=(n2)tan^n^^3xdx,,,,,,v=tanx $
$\displaystyle uv\int vdu $
$\displaystyle tan^n^^1x(n2)\int tan^n^^2xdx  \int tan^n^^2xdx$
so I get
$\displaystyle \int tan^nx = tan^n^^1x(n1)\int tan^n^^2xdx $ Where's did I go wrong?

The derivative of $\displaystyle \tan^{n2} x$ is not $\displaystyle (n2) \tan^{n3} x$. Do you see why?

Oh yeah!
Yeah, the Chain Rule. Thanks for the backup