A Step Involving the Binomial Theorem: Can You Explain Why its Valid?

A Step Involving the Binomial Theorem: Can You Explain Why its Valid?

I have been given a step in an evaluation of an integral, but I can't work
out what theorem has been used:
See:
$=\int_d^e\sum\limits_{n=0}^\infty\dfrac{(-1)^na^n\left((b+c)\cos
x-\sqrt{b^2-(b+c)^2\sin^2x}\right)^{2n}}{n!}dx$
$=\int_d^e\sum\limits_{n=0}^\infty\sum\limits_{m=0}^n\dfrac{(-1)^nC_{2m}^{2n}a^n(b+c)^{2n-2m}\cos^{2n-2m}x\left(\sqrt{b^2-(b+c)^2\sin^2x}\right)^{2m}}{n!}dx-\int_d^e\sum\limits_{n=0}^\infty\sum\limits_{m=1}^n\dfrac{(-1)^nC_{2m-1}^{2n}a^n(b+c)^{2n-2m+1}\cos^{2n-2m+1}x\left(\sqrt{b^2-(b+c)^2\sin^2x}\right)^{2m-1}}{n!}dx$
Certainly the binomial theorem has been used, but why are the two terms
valid?
There must be a trick used but I can't work it out...
-Alex