My Ferjer kernel is defined to be $F_N(t):= \frac{1}{N+1}(\frac{sin(N+1)\pi t}{sin \pi t})^2$, I want to show that $F_N(t) \leq \frac{c}{N+1}min\{N+1, \frac{1}{x^2}\}$ on $[\frac{1}{-2}, \frac{1}{2}]$.
Thoughts: I need some kind of bound on $sin(N+1)\pi t$, but I do not see how to derive a useful bound.
Each different book normalizes the Fourier coefficient in a different way, here is a general idea to achieve your goal. You could change the normalization, i.e. dividing some $\pi$, or $2\pi$, or translation, etc.
Firstly, $\sin n\alpha\leq n\sin\alpha$. Thus, \begin{align*} |F_{N}(x)|&=\dfrac{1}{N+1}\dfrac{|\sin\frac{N+1}{2}x|^{2}}{|\sin\frac{x}{2}|^{2}}\\ &\leq\dfrac{1}{N+1}(N+1)^{2}\dfrac{|\sin\frac{x}{2}|^{2}}{|\sin\frac{x}{2}|^{2}} \\ &=N+1, \end{align*} so you have your fist bound.
For the other one, you need to know the fact that $$|\sin\frac{x}{2}|\geq\dfrac{|x|}{\pi},\ \text{for}\ 0<|x|<\pi.$$
Thus, \begin{align*} |F_{N}(x)|&=\dfrac{1}{N+1}\dfrac{|\sin\frac{N+1}{2}x|^{2}}{|\sin\frac{x}{2}|^{2}}\\ &\leq\dfrac{1}{N+1}\dfrac{1}{|\sin\frac{x}{2}|^{2}} \\ &\leq \dfrac{\pi^{2}}{(N+1)x^{2}}. \end{align*}
This gives you the second bound.
Depending on the normalization, some books will normalize the kernel by dividing either $\pi$ or $2\pi$, but you have the constant $C$, so no need to worry.