Question

\(\text{ cos(x+\frac{\Pi}{3})+cosx=\frac{3}{2}-4sin\frac{x}{2}.sin(\frac{x}{2}+\frac{\Pi}{6})}\)

Answer 1

\(\Leftrightarrow2cos\left(x+\frac{\pi}{6}\right).cos\left(\frac{\pi}{6}\right)=\frac{3}{2}-4sin\frac{x}{2}.sin\left(\frac{x}{2}+\frac{\pi}{6}\right)\)

\(\Leftrightarrow\sqrt{3}cos\left(x+\frac{\pi}{6}\right)=\frac{3}{2}+2\left[cos\left(x+\frac{\pi}{6}\right)-cos\frac{\pi}{6}\right]\)

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

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

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

\(\Leftrightarrow...\)