\(tana+tanb=\frac{sina.cosb+cosa.sinb}{cosa.cosb}=\frac{sin\left(a+b\right)}{cosa.cosb}\)
\(tana-tanb=\frac{sina.cosb-cosa.sinb}{cosa.cosb}=\frac{sin\left(a-b\right)}{cosa.cosb}\)
\(tan\left(\frac{\pi}{3}-3x\right)-tan\left(\frac{\pi}{3}\right)+tan2x+tanx=0\)
\(\Leftrightarrow\frac{-sin3x}{cos\left(\frac{\pi}{3}-3x\right).cos\left(\frac{\pi}{3}\right)}+\frac{sin3x}{cosx.cos2x}=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sin3x=0\\cosx.cos2x=\frac{1}{2}cos\left(\frac{\pi}{3}-3x\right)\end{matrix}\right.\)
Pt dưới \(\Leftrightarrow cos3x+cosx=cos\left(\frac{\pi}{3}-3x\right)\)
\(\Leftrightarrow cos3x-cos\left(\frac{\pi}{3}-3x\right)+cosx=0\)
\(\Leftrightarrow-2sin\left(\frac{\pi}{6}\right).sin\left(3x-\frac{\pi}{6}\right)+cosx=0\)
\(\Leftrightarrow sin\left(3x-\frac{\pi}{6}\right)=-cosx=sin\left(x-\frac{\pi}{2}\right)\)
