# Math Help - [SOLVED] Application of Binomial theorem?

1. ## [SOLVED] Application of Binomial theorem?

Prove that:
$\sum\limits_{r = 0}^{n} {{n\choose r}\frac{(\cos 2x)^{r + 1}}{r + 1}} = \frac{2^{n + 1}{\cos}^{2(n + 1)}x - 1}{n + 1}$

2. Originally Posted by fardeen_gen
Prove that:
$\sum\limits_{r = 0}^{n} {{n\choose r}\frac{(\cos 2x)^{r + 1}}{r + 1}} = \frac{2^{n + 1}{\cos}^{2(n + 1)}x - 1}{n + 1}$
$2\cos^2(x)-1=\cos(2x)$

so:

$2^{n+1}\cos^{2(n+1)}(x)=(\cos(2x)+1)^{n+1}$

Use the binomial expansion of the right hand side above to show that:

$(\cos(2x)+1)^{n+1}=1+(n+1)\left[\sum\limits_{r = 0}^{n} {{n\choose r}\frac{(\cos 2x)^{r + 1}}{r + 1}} \right]$

CB