c)  $\sin 4x-\text{cos}\left(x+\frac{\pi }{6}\right)=0$.

c)  $\sin 4x-\text{cos}\left(x+\frac{\pi }{6}\right)=0$

 $\iff \sin 4x=\text{cos}\left(x+\frac{\pi }{6}\right)$

 $\iff \sin \left(4x\right)=\text{cos}\left(x+\frac{2\pi }{3}-\frac{\pi }{2}\right)$

  $\iff \sin \left(4x\right)=\text{cos}\left(\frac{\pi }{2}-\left(x+\frac{2\pi }{3}\right)\right)$

 $\iff \sin \left(4x\right)=\sin \left(x+\frac{2\pi }{3}\right)$

 $\iff \left[\begin{array}{l}4x=x+\frac{2\pi }{3}+k2\pi \\ 4x=-x-\frac{2\pi }{3}+k2\pi \end{array}\right.,k\in \mathbb{Z}$

$\iff \left[\begin{array}{l}x=\frac{2\pi }{9}+k\frac{2\pi }{3}\\ x=-\frac{2\pi }{15}+k\frac{2\pi }{5}\end{array}\right.,k\in \mathbb{Z}$

Vậy tập nghiệm của phương trình là:  $S=\left\{\frac{2\pi }{9}+k\frac{2\pi }{3};-\frac{2\pi }{15}+k\frac{2\pi }{5},k\in \mathbb{Z}\right\}$.