Question

GPT: cosx + cos2x = sin3x

Answer 1

\(\Leftrightarrow2cos\frac{3x}{2}.cos\frac{x}{2}=2sin\frac{3x}{2}.cos\frac{3x}{2}\)

\(\Leftrightarrow\left[{}\begin{matrix}cos\frac{3x}{2}=0\\cos\frac{x}{2}=sin\frac{3x}{2}=cos\left(\frac{\pi}{2}-3x\right)\end{matrix}\right.\)

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

\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{3}+\frac{k2\pi}{3}\\x=\frac{\pi}{7}+\frac{k4\pi}{7}\\x=\frac{\pi}{5}+\frac{k4\pi}{5}\end{matrix}\right.\)