Question

\(4\sin^{2020}x+4\cos^{2020}x=8\left(sin^{2022}x+\cos^{2022}x\right)+5\cos2x\)

Giải pt

 

Answer 1

\(\Leftrightarrow4sin^{2020}x\left(1-2sin^2x\right)=4cos^{2020}x\left(2cos^2x-1\right)+5cos2x=0\)

\(\Leftrightarrow4sin^{2020}x.cos2x=4cos^{2020}x.cos2x+5cos2x\)

\(\Leftrightarrow\left[{}\begin{matrix}cos2x=0\Rightarrow x=...\\4sin^{2020}x=4cos^{2020}x+5\left(1\right)\end{matrix}\right.\)

Xét (1), ta có \(\left\{{}\begin{matrix}4sin^{2020}x\le4\\4cos^{2020}x+5\ge5\end{matrix}\right.\)

\(\Rightarrow4sin^{2020}x< 4cos^{2020}x+5\) với mọi x

\(\Rightarrow\left(1\right)\) vô nghiệm