9.
\(cos\left(-3x-\dfrac{\pi}{4}\right)+sin\left(x-\dfrac{\pi}{4}\right)=0\)
\(\Leftrightarrow sin\left(3x+\dfrac{\pi}{4}\right)+sin\left(x-\dfrac{\pi}{4}\right)=0\)
\(\Leftrightarrow2sin2x.cos\left(x+\dfrac{\pi}{8}\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sin2x=0\\cos\left(x+\dfrac{\pi}{8}\right)=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}2x=k\pi\\x+\dfrac{\pi}{8}=\dfrac{\pi}{2}+k\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{k\pi}{2}\\x=\dfrac{3\pi}{8}+k\pi\end{matrix}\right.\)
10.
\(sin\left(x+\dfrac{\pi}{6}\right)+cos\left(5x+\dfrac{\pi}{6}\right)=0\)
\(\Leftrightarrow sin\left(x+\dfrac{\pi}{6}\right)+sin\left(\dfrac{\pi}{3}-5x\right)=0\)
\(\Leftrightarrow2sin\left(-2x+\dfrac{\pi}{4}\right).cos\left(\dfrac{3x}{2}-\dfrac{\pi}{12}\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sin\left(-2x+\dfrac{\pi}{4}\right)=0\\cos\left(\dfrac{3x}{2}-\dfrac{\pi}{12}\right)=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}-2x+\dfrac{\pi}{4}=k\pi\\\dfrac{3x}{2}-\dfrac{\pi}{12}=\dfrac{\pi}{2}+k\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{8}-\dfrac{k\pi}{2}\\x=\dfrac{7\pi}{18}+\dfrac{k2\pi}{3}\end{matrix}\right.\)
11.
ĐK: \(x\ne\dfrac{\pi}{12}+\dfrac{k\pi}{2};x\ne\dfrac{k\pi}{2}\)
\(tan\left(2x+\dfrac{\pi}{3}\right)+cot3x=0\)
\(\Leftrightarrow tan\left(2x+\dfrac{\pi}{3}\right)=tan\left(3x-\dfrac{\pi}{2}\right)\)
\(\Leftrightarrow2x+\dfrac{\pi}{3}=3x-\dfrac{\pi}{2}+k\pi\)
\(\Leftrightarrow x=\dfrac{5\pi}{6}+k\pi\)
