a: \(cos\left(4x+\dfrac{\Omega}{3}\right)+sin\left(x-\dfrac{\Omega}{4}\right)=0\)
=>\(cos\left(4x+\dfrac{\Omega}{3}\right)=-sin\left(x-\dfrac{\Omega}{4}\right)=sin\left(-x+\dfrac{\Omega}{4}\right)\)
=>\(cos\left(4x+\dfrac{\Omega}{3}\right)=cos\left(\dfrac{\Omega}{2}+x-\dfrac{\Omega}{4}\right)=cos\left(x+\dfrac{\Omega}{4}\right)\)
=>\(\left[{}\begin{matrix}4x+\dfrac{\Omega}{3}=-x-\dfrac{\Omega}{4}+k2\Omega\\4x+\dfrac{\Omega}{3}=x+\dfrac{\Omega}{4}+k2\Omega\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}5x=-\dfrac{7}{12}\Omega+k2\Omega\\3x=-\dfrac{\Omega}{12}+k2\Omega\end{matrix}\right.\)
=>\(\left[{}\begin{matrix}x=-\dfrac{7}{60}\Omega+\dfrac{k2\Omega}{5}\\x=-\dfrac{\Omega}{36}+\dfrac{k2\Omega}{3}\end{matrix}\right.\)
b: \(cos^2\left(3x+\dfrac{\Omega}{3}\right)=cos^2x\)
=>\(\left[{}\begin{matrix}3x+\dfrac{\Omega}{3}=x+k2\Omega\\3x+\dfrac{\Omega}{3}=-x+k2\Omega\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}2x=-\dfrac{\Omega}{3}+k2\Omega\\4x=-\dfrac{\Omega}{3}+k2\Omega\end{matrix}\right.\)
=>\(\left[{}\begin{matrix}x=-\dfrac{\Omega}{6}+k\Omega\\x=-\dfrac{\Omega}{12}+\dfrac{k\Omega}{2}\end{matrix}\right.\Leftrightarrow x=-\dfrac{\Omega}{12}+\dfrac{k\Omega}{2}\)
