The question is prove that $\lim_{N \to \infty} \int_{\pi-\pi/(N+1/2)}^{\pi} \frac{\sin((N+1/2)t)}{\sin(t/2)} dt = 0$.
I can show that $$\int_{\pi-\pi/(N+1/2)}^{\pi} \frac{\sin((N+1/2)t)}{\sin(t/2)} dt = \int_{\pi-\pi/(N+1/2)}^{\pi} \sin(N t) \frac{\cos(t/2)}{\sin(t/2)} dt + \int_{\pi-\pi/(N+1/2)}^{\pi} \cos(N t) dt$$ and that $ \int_{\pi-\pi/(N+1/2)}^{\pi} \cos(N t) dt$ has limit 0, but working with $\int_{\pi-\pi/(N+1/2)}^{\pi} \sin(N t) \frac{\cos(t/2)}{\sin(t/2)} dt$ is a little harder.
I do know that, for any fixed $M$, $\int_{\pi-\pi/(M+1/2)}^{\pi} \sin(N t) \frac{\cos(t/2)}{\sin(t/2)} dt$ has limit equal to 0 when $N \to \infty$ using some analysis relating to Fourier Coefficients, but proving this result with a moving endpoint is a little more difficult.