Question

A = \(\dfrac{\left(cos10x+cos7x\right)-\left(cos9x+cos8x\right)}{\left(sin10x+sin7x\right)-\left(sin9x+sin8x\right)}\) = cot\(\dfrac{m}{n}x\), với m,n là phân số tối giản.Tính m + n

Answer 1

\(A=\dfrac{\left(cos10x-cos8x\right)-\left(cos9x-cos7x\right)}{\left(sin10x-sin8x\right)-\left(sin9x-sin7x\right)}\)

\(=\dfrac{-2sin9x.sinx+2sin8x.sinx}{2cos9x.sinx-2cos8x.sinx}=\dfrac{2sinx\left(-sin9x+sin8x\right)}{2sinx\left(cos9x-cos8x\right)}\)

\(=\dfrac{sin8x-sin9x}{cos9x-cos8x}=\dfrac{-2cos\left(\dfrac{17x}{2}\right)sin\left(\dfrac{x}{2}\right)}{-2sin\left(\dfrac{17x}{2}\right)sin\left(\dfrac{x}{2}\right)}\)

\(=cot\left(\dfrac{17}{2}x\right)\)

\(m+n=17+2=19\)