Biểu thức \(A = \frac{{\left( {\cos 10x + \cos 7x} \right) - \left( {\cos 9x + \cos 8x} \right)}}{{\left( {\sin 10x + \sin 7x} \right) - \left( {\sin 9x + \sin 8x} \right)}} = \cot \frac{m}{n}x\) với \(\frac{m}{n}\) là phân số tối giản. Tính m + n.

\(A = \frac{{\left( {\cos 10x + \cos 7x} \right) - \left( {\cos 9x + \cos 8x} \right)}}{{\left( {\sin 10x + \sin 7x} \right) - \left( {\sin 9x + \sin 8x} \right)}}\)

\( = \frac{{2\cos \frac{{17x}}{2}\cos \frac{{3x}}{2} - 2\cos \frac{{17x}}{2}\cos \frac{x}{2}}}{{2\sin \frac{{17x}}{2}\cos \frac{{3x}}{2} - 2\sin \frac{{17x}}{2}\cos \frac{x}{2}}}\)\( = \frac{{2\cos \frac{{17x}}{2}\left( {\cos \frac{{3x}}{2} - \cos \frac{x}{2}} \right)}}{{2\sin \frac{{17x}}{2}\left( {\cos \frac{{3x}}{2} - \cos \frac{x}{2}} \right)}}\)\( = \frac{{cos\frac{{17x}}{2}}}{{\sin \frac{{17x}}{2}}}\)\( = \cot \frac{{17x}}{2}\).

Suy ra m = 17; n = 2. Suy ra m + n = 19.

Trả lời: 19.