Recall that [tex]\cos(x)+\cos(y)=2\cos \left(\frac{x+y}{2} \right)\cos \left(\frac{x-y}{2} \right)[/tex].
[tex]\cos(12x)+\cos(16x)=0 \\ \\ 2\cos(2x)\cos(14x)=0 \\ \\ \cos(2x)\cos(14x)=0[/tex]
If [tex]\cos(2x)=0[/tex], then [tex]x=\frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \cdots[/tex].
If [tex]\cos(14x)=0[/tex], then [tex]x=\frac{\pi}{28}, \frac{3\pi}{28}, \frac{5\pi}{28}, \cdots[/tex].
So, the three smallest positive [tex]x[/tex] intercepts are [tex]\frac{\pi}{28}, \frac{3\pi}{28}, \frac{5\pi}{28}[/tex].
