Giải phương trình: sin^3xcos3x+cos^3xsin3x=sin^34x

Hướng dẫn giải:

Ta có: (*)\(\Leftrightarrow {{\sin }^{3}}x\left( 4{{\cos }^{3}}x-3\cos x \right)+{{\cos }^{3}}x\left( 3\sin x-4{{\sin }^{3}}x \right)=si{{n}^{3}}4x\)

 \( \Leftrightarrow 4{{\sin }^{3}}xco{{s}^{3}}x-3{{\sin }^{3}}xcosx+3sinxco{{s}^{3}}x-4si{{n}^{3}}xco{{s}^{3}}x=si{{n}^{3}}4x \)

\( \Leftrightarrow 3\sin x\cos x\left( {{\cos }^{2}}x-si{{n}^{2}}x \right)={{\sin }^{3}}4x\Leftrightarrow \frac{3}{2}\sin 2x\cos 2x={{\sin }^{3}}4x \)

 \( \Leftrightarrow \frac{3}{4}\sin 4x={{\sin }^{3}}4x\Leftrightarrow 3\sin 4x-4{{\sin }^{3}}4x=0\Leftrightarrow \sin 12x=0 \)

 \( \Leftrightarrow 12x=k\pi \Leftrightarrow x=\frac{k\pi }{12},\text{ }k\in \mathbb{Z} \).