Showing $2x=\left( 2n+1\right) \pi \left( 1-\cos x\right) $ has $2n+3$ roots when $n\in \mathbb{Z}_+$ I am struggling to show that the equation $$2x=\left( 2n+1\right) \pi \left( 1-\cos x\right) $$ where n is a positive integer, has $2n+3$ roots and no more and also if it possible to indicate their locations (even roughly) ? 
I am unsure how proceed with this one due to the presence of both $\cos x$ and $2x$ and would be great full if some one could give me a hint.
Cheers

Hint: The equation easily transforms to $x=(2n+1)\frac{\pi}{2}\sin^{2}x$. It is easy to see that there are no roots for $x&lt0$ and $x>(2n+1)\frac{\pi}2$. Also, $(2n+1)\frac{\pi}{2}\sin^{2}x=0$ at $x= k\pi$.
