1.
$\sin ^2 \Big(3x-\dfrac{\pi}{2}\Big)= \dfrac{3}{4}$
$\Leftrightarrow \dfrac{1-\cos (6x-\pi)}{2}=\dfrac{3}{4}$
$\Leftrightarrow 1+ \cos (2\pi - 6x)=\dfrac{3}{2}$
$\Leftrightarrow \cos (2\pi - 6x)=\dfrac{1}{2}$
$\Leftrightarrow \left[ \begin{array}{l}2\pi - 6x=\dfrac{\pi}{3}+k2\pi\\2\pi - 6x=\dfrac{-\pi}{3}+k2\pi \end{array}\right.$
$\Leftrightarrow \left[ \begin{array}{l}x=\dfrac{5\pi}{18}-\dfrac{k\pi}{3}\\x=\dfrac{7\pi}{18}-\dfrac{k\pi}{3} \end{array} \right.(k \in \mathbb{Z})$
2.
$\cos x + \cos 2x + \cos 3x + \cos 4x =0$
$\Leftrightarrow ( \cos x + \cos 3x) + (\cos 2x + \cos 4x)=0$
$\Leftrightarrow \Big( 2 \cos \dfrac{4x}{2}.\cos \dfrac{-2x}{2}\Big)+ \Big( 2\cos \dfrac{6x}{2}.\cos \dfrac{-2x}{2}\Big)=0$
$\Leftrightarrow 2 \cos (-x). (\cos 2x +\cos 3x) =0$
$\Leftrightarrow \left[ \begin{array}{l}\cos x=0\\\cos 2x =- \cos 3x \end{array} \right.$
$\Leftrightarrow \left[ \begin{array}{l}x=\dfrac{\pi}{2}+k\pi\\x=\dfrac{\pi}{3}+\dfrac{k2\pi}{3}\\x=\pi - k2\pi \end{array} \right.(k \in \mathbb{Z})$
3.
