I'm having some trouble showing this proof for the reduction formula for $\int \cos^{n}{x}\,dx$. I already know how to prove the regular reduction formula for $\int \cos^{n}{x}\,dx$:
$$\int \cos^n{x}\,dx=\frac{{1}}n(\sin{x})(\cos^{n-1}{x})+\frac{{n-1}}n\int \cos^{n-2}\,{x}\,dx$$
What I'm trying to prove is (what seems to be) an alternate reduction formula provided by my professor:
$$\int \cos^n{x}\,dx=\frac{{1}}n(\sin{x})(\cos^{n-1}{x})+\;\pmb{\left[\frac{{n}}{n-2}\right]}\;\int \cos^{n-2}\,{x}\,dx$$
Does anyone know if this second proof for the reduction formula for $\int \cos^{n}{x}\,dx$ is actually true, and if it is, can you show how to prove it...? If it's not, then I can let my professor know that that this proof is erroneous...
Thanks for the help.