40.
\(cos^2\left(2x-\dfrac{\pi}{4}\right)=\dfrac{3}{4}\)
\(\Leftrightarrow\left[{}\begin{matrix}cos\left(2x-\dfrac{\pi}{4}\right)=\dfrac{\sqrt{3}}{2}\\cos\left(2x-\dfrac{\pi}{4}\right)=-\dfrac{\sqrt{3}}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}2x-\dfrac{\pi}{4}=\pm\dfrac{\pi}{6}+k2\pi\\2x-\dfrac{\pi}{4}=\pm\dfrac{5\pi}{6}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{5\pi}{24}+k\pi\\x=\dfrac{\pi}{24}+k\pi\\x=\dfrac{13\pi}{24}+k\pi\\x=-\dfrac{7\pi}{24}+k\pi\end{matrix}\right.\)
42.
\(cos\left(3x+\dfrac{\pi}{3}\right)+sin\left(\dfrac{5\pi}{6}+3x\right)=2\)
\(\Leftrightarrow cos\left(3x+\dfrac{\pi}{3}\right)+cos\left(-\dfrac{\pi}{3}-3x\right)=2\)
\(\Leftrightarrow2cos0.cos\left(6x+\dfrac{2\pi}{3}\right)=2\)
\(\Leftrightarrow cos\left(6x+\dfrac{2\pi}{3}\right)=1\)
\(\Leftrightarrow6x+\dfrac{2\pi}{3}=k2\pi\)
\(\Leftrightarrow x=-\dfrac{\pi}{9}+\dfrac{k\pi}{3}\)
41.
\(cos\left(4x+\dfrac{\pi}{5}\right)-sin2x=0\)
\(\Leftrightarrow cos\left(4x+\dfrac{\pi}{5}\right)=cos\left(\dfrac{\pi}{2}-2x\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}4x+\dfrac{\pi}{5}=\dfrac{\pi}{2}-2x+k2\pi\\4x+\dfrac{\pi}{5}=-\dfrac{\pi}{2}+2x+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{20}+\dfrac{k\pi}{3}\\x=\dfrac{7\pi}{20}+k\pi\end{matrix}\right.\)
