Giải phương trình: cosx+cos2x+cos3x+cos4x=0

Hướng dẫn giải:

Ta có: (*)  \( \Leftrightarrow (\cos x+\cos 3x)+(\cos 2x+\cos 4x)=0\Leftrightarrow 2\cos 2x.\cos x+2\cos 3x.\cos x=0 \)

 \( \Leftrightarrow 2\cos x(\cos 3x+\cos x)=0\Leftrightarrow 4\cos x.\cos \frac{5x}{2}.\cos \frac{x}{2}=0\Leftrightarrow \left[ \begin{align}  & \cos x=0 \\  & \cos \frac{5\pi }{2}=0 \\  & \cos \frac{x}{2}=0 \\ \end{align} \right. \)

\(\Leftrightarrow \left[ \begin{align}  & x=\frac{\pi }{2}+k\pi  \\  & \frac{x}{2}=\frac{\pi }{2}+k\pi  \\  & \frac{5x}{2}=\frac{\pi }{2}+k\pi  \\ \end{align} \right.\)\(\Leftrightarrow \left[ \begin{align}  & x=\frac{\pi }{2}+k\pi  \\  & x=\pi +k2\pi  \\  & x=\frac{\pi }{5}+\frac{k2\pi }{5} \\ \end{align} \right.,\text{ }(k\in \mathbb{Z})\).