Rút gọn biểu thức A=2cos^2(2alpha)+căn(3)sin(4alpha)−1/2sin^2(2alpha)+căn(3)sin(4alpha)−1 .

Ta có \(A = \frac{{2{{\cos }^2}2\alpha + \sqrt 3 \sin 4\alpha - 1}}{{2{{\sin }^2}2\alpha + \sqrt 3 \sin 4\alpha - 1}}\)

\( = \frac{{\cos 4\alpha + \sqrt 3 \sin 4\alpha }}{{\sqrt 3 \sin 4\alpha - \cos 4\alpha }}\)

\[ = \frac{{\frac{1}{2}\cos 4\alpha + \frac{{\sqrt 3 }}{2}\sin 4\alpha }}{{\frac{{\sqrt 3 }}{2}\sin 4\alpha - \frac{1}{2}\cos 4\alpha }}\]

\[ = \frac{{\sin \left( {4\alpha + 30^\circ } \right)}}{{\sin \left( {4\alpha - 30^\circ } \right)}}\].