Question

\(\frac{c\text{os}\left(x+\frac{5\pi}{6}\right)}{c\text{os}\left(2x-\frac{\pi}{6}\right)}+tan\left(2x-\frac{\pi}{6}\right)=0\)

Answer 1

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\(\Leftrightarrow\frac{cos\left(x+\frac{5\pi}{6}\right)}{cos\left(2x-\frac{\pi}{6}\right)}+\frac{sin\left(2x-\frac{\pi}{6}\right)}{cos\left(2x-\frac{\pi}{6}\right)}=0\)

\(\Leftrightarrow cos\left(x+\frac{5\pi}{6}\right)+sin\left(2x-\frac{\pi}{6}\right)=0\)

\(\Leftrightarrow cos\left(x+\frac{5\pi}{6}\right)=-sin\left(2x-\frac{\pi}{6}\right)\)

\(\Leftrightarrow cos\left(x+\frac{5\pi}{6}\right)=cos\left(2x+\frac{\pi}{3}\right)\)

\(\Leftrightarrow\left[{}\begin{matrix}2x+\frac{\pi}{3}=x+\frac{5\pi}{6}+k2\pi\\2x+\frac{\pi}{3}=-x-\frac{5\pi}{6}+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{2}+k2\pi\\x=-\frac{7\pi}{18}+\frac{k2\pi}{3}\end{matrix}\right.\)