Question

Giải phương trình: \(\left(2\cos2x-1\right)\cos x-\sin x=\sqrt{2}\left(\sin x+\cos x\right)\sin3x\)

Answer 1

\(\Leftrightarrow2cosx.cos2x-\left(cosx+sinx\right)-\sqrt{2}sin3x\left(cosx+sinx\right)=0\)

\(\Leftrightarrow2cosx\left(cos^2x-sin^2x\right)-\left(cosx+sinx\right)\left(1+\sqrt{2}sin3x\right)=0\)

\(\Leftrightarrow\left(cosx+sinx\right)\left(2cos^2x-2sinx.cosx\right)-\left(cosx+sinx\right)\left(1+\sqrt{2}sin3x\right)=0\)

\(\Leftrightarrow\left(cosx+sinx\right)\left(2cos^2x-sin2x-1-\sqrt{2}sin3x\right)=0\)

Biến đổi ngoặc sau:

\(cos2x-sin2x=\sqrt{2}sin3x\)

\(\Leftrightarrow-\sqrt{2}sin\left(2x-\frac{\pi}{4}\right)=\sqrt{2}sin3x\)

\(\Leftrightarrow sin\left(\frac{\pi}{4}-2x\right)=sin3x\)