Trigonometric series problem I have the following problem from my Fourier analysis book I would need some guidance with. I have tried it, but apparently I made some mistakes...here is my problem: 

We have: 
$$\sin \theta = \frac{2}{\pi}-\frac{4}{\pi}\sum_1^{\infty}\frac{\cos 2n\theta}{4n^2-1}\;\;\;\;(0\leq\theta\leq \pi)\;\;\;\;(*),$$ and we
  also have: 
$$\cos\theta=\frac{d}{d\theta}\sin\theta = -\int_{\pi/2}^\theta \sin\phi \:d\phi.$$
Show that series $(*)$ can be differentiated and integrated termwise
  to yield two apparently different expressions for $\cos\theta$ for $0 <\theta < \pi$, and reconcile these two expressions. (Hint the
  following equation is useful)
$$\sum_1^{\infty}\frac{(-1)^{n+1}}{n}\sin n\theta = \frac{\theta}{2}\;\;\text{for}\;\;-\pi < \theta < \pi.$$

Could someone give me some guidelines to this problem? =) I tried first differentiating $(*)$ and then integrating $(*)$, but I'm not sure if I got it right or not...
thnx for any guidance =)
My attempted solution: 
$$\frac{d}{d\theta}\sin\theta = \frac{8}{\pi}\sum_1^{\infty}\frac{n}{4n^2-1}\sin2n\theta = \cos\theta$$
$$\cos\theta = -\int_{\pi/2}^\theta \sin\phi \;d\phi = -\int_{\pi/2}^\theta \left[\frac{2}{\pi}-\frac{4}{\pi}\sum_1^{\infty}\frac{\cos 2n\phi}{4n^2-1}\right]\;d\phi$$
$$= -\int_{\pi/2}^\theta \frac{2}{\pi}\;d\phi + \frac{4}{\pi}\int_{\pi/2}^\theta\sum_1^{\infty}\frac{\cos 2n\phi}{4n^2-1}\;d\phi = -\left[ \frac{2\phi}{\pi} \right]_{\pi/2}^{\theta} + \frac{4}{\pi}\left[ \sum_1^{\infty}\frac{\sin2n\phi}{2n(4n^2-1)}\right]_{\pi/2}^\theta $$
$$= -\left[\frac{2\theta}{\pi}-1\right] + \frac{4}{\pi}\left[ \sum_1^{\infty}\frac{\sin2n\theta}{2n(4n^2-1)}- \sum_1^{\infty}\frac{\sin n\pi}{2n(4n^2-1)} \right]$$
$$= 1-\frac{2\theta}{\pi} + \frac{4}{\pi}\sum_1^{\infty}\frac{\sin2n\theta}{2n(4n^2-1)} = \cos\theta \;\;\;\text{?} $$
Here is where I get stuck...
So does: 
$$1-\frac{2\theta}{\pi} + \frac{4}{\pi}\sum_1^{\infty}\frac{\sin2n\theta}{2n(4n^2-1)} = \frac{8}{\pi}\sum_1^{\infty}\frac{n}{4n^2-1}\sin2n\theta = \cos\theta\;\;\;\;\;\;\;\; 0<\theta<\pi\;\;\;\;\text{?}$$
UPDATE : 
It seems my calculations were correct, I tried visualizing my series in Matlab:


Now I just need to prove that the Fourier series are equal to each other =) 

Hints:
First of all:
$$\left|\frac{\cos 2n\theta}{4n^2-1}\right|\le\frac1{4n^2-1}\implies \;\text{since the series} \sum_{n=1}^\infty\frac1{4n^2-1}$$
converges, Weierstrass $\;M$-test gives so uniform convergence for all $\;\theta\in\Bbb R\;$ , which means we can differentiate-integrate termwise.
Differentiation:
$$(\sin\theta)'=\cos\theta\stackrel?=\left(\frac{2}{\pi}-\frac{4}{\pi}\sum_1^{\infty}\frac{\cos 2n\theta}{4n^2-1}\right)'=\frac4\pi\sum_{n=1}^\infty\frac{2n\sin2n\theta}{4n^2-1}$$
Integration:
$$\int\sin\theta d\theta=-\cos\theta\stackrel?=\int\left(\frac{2}{\pi}-\frac{4}{\pi}\sum_1^{\infty}\frac{\cos 2n\theta}{4n^2-1}\right)d\theta=$$
$$=\frac2\pi\theta-\frac4\pi\sum_{n=1}^\infty\frac{\sin2n\theta}{2n(4n^2-1)}$$
