Problem rearranging a trigonometric expression I've been trying to figure out why this equation is satisfied:
$$\frac{1-(\cos(x))^3}{x^2}=\frac{2\cdot(\sin(\frac{x}{2}))^2}{x^2}\cdot(1+\cos(x)+(\cos(x))^2)$$
but I can't find the proper trigonometric changes in order to change from one to another. I know that the sine comes from the double angle formula, but I obtain slightly different results in other parts and it's never the same as the formula above.
I'd appreciate any hint.
Thanks for the time.

One may observe that, in general,
$$
a^3-b^3=(a-b)(a^2+ab+b^2)
$$ giving
$$
1-(\cos x)^3=(1-\cos x)(1+\cos x+(\cos x)^2)
$$ then, by using $2\sin^2(x/2)=1-\cos x$ as noticed by @André Nicolas, you are Ok with your expression.
