Question

Giải phương trình 

cos2x + cosx + 1= sin2x+sinx

Answer 1

\(cos2x+cosx+1=sin2x+sinx\)

\(\Leftrightarrow cos^2x-sin^2x+cosx+cos^2x+sin^2x=2sinx.cosx+sinx\)

\(\Leftrightarrow2cos^2x+cosx=2sinx.cosx+sinx\)

\(\Leftrightarrow cosx\left(2cosx+1\right)=sinx\left(2cosx+1\right)\)

\(\Leftrightarrow\left(2cosx+1\right)\left(sinx-cosx\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}2cosx+1=0\\sinx=cosx\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}cosx=-\dfrac{1}{2}\\tanx=1\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x=\pm\dfrac{\pi}{3}+k2\pi\\x=\dfrac{\pi}{4}+k\pi\\\end{matrix}\right.\)