Rút gọn biểu thức sau: A = (sin x + sin 2 x + sin 3 x + sin 4 x)/( cos x + cos 2 x + cos 3 x + cos 4 x)

\(\begin{array}{*{20}{l}}A&{ = \frac{{\sin x + \sin 2x + \sin 3x + \sin 4x}}{{\cos x + \cos 2x + \cos 3x + \cos 4x}}}\\{}&{ = \frac{{(\sin 3x + \sin x) + (\sin 4x + \sin 2x)}}{{(\cos 3x + \cos x) + (\cos 4x + \cos 2x)}} = \frac{{2\sin 2x\cos x + 2\sin 3x\cos x}}{{2\cos 2x\cos x + 2\cos 3x\cos x}}}\\{}&{ = \frac{{2\cos x(\sin 2x + \sin 3x)}}{{2\cos x(\cos 2x + \cos 3x)}} = \frac{{2\sin \frac{{5x}}{2}\cos \frac{x}{2}}}{{2\cos \frac{{5x}}{2}\cos \frac{x}{2}}} = \tan \frac{{5x}}{2}}\end{array}\)