Giải phương trình: cos^4x+sin^4x+cos(x−π/4)sin(3x−π/4)−3/2=0

Hướng dẫn giải:

Ta có: (*) \( \Leftrightarrow {{({{\sin }^{2}}x+co{{s}^{2}}x)}^{2}}-2{{\sin }^{2}}xco{{s}^{2}}x+\frac{1}{2}\left[ \sin \left( 4x-\frac{\pi }{2} \right)+\sin 2x \right]-\frac{3}{2}=0 \)

 \( \Leftrightarrow 1-\frac{1}{2}{{\sin }^{2}}2x+\frac{1}{2}[-\cos 4x+\sin 2x]-\frac{3}{2}=0 \)

 \( \Leftrightarrow -\frac{1}{2}{{\sin }^{2}}2x-\frac{1}{2}(1-2{{\sin }^{2}}2x)+\frac{1}{2}\sin 2x-\frac{1}{2}=0\Leftrightarrow {{\sin }^{2}}2x+sin2x-2=0 \)

 \( \Leftrightarrow \left[ \begin{align}  & \sin 2x=1\text{ }(n) \\  & \sin 2x=-2\text{ }(\ell ) \\ \end{align} \right.\Leftrightarrow 2x=\frac{\pi }{2}+k2\pi \Leftrightarrow x=\frac{\pi }{4}+k2\pi ,\text{ }k\in \mathbb{Z} \).