Question

Chứng minh đẳng thức:

\(\sin14x+\sin7x-\sin5x-\sin16x=4\sin\dfrac{9x}{2}\sin\dfrac{21x}{2}\sin x\)

Answer 1

Ta co': \(VT=sin14x+sin7x-\left(sin5x+sin16x\right)\)
                  \(=2.sin\dfrac{21x}{2}.cos\dfrac{7x}{2}-\left(2.sin\dfrac{21x}{2}.cos\dfrac{-11x}{2}\right)\)
                  \(=2.sin\dfrac{21x}{2}.cos\dfrac{7x}{2}-2.sin\dfrac{21x}{2}.cos\dfrac{11x}{2}\)
                  \(=2.sin\dfrac{21x}{2}.\left(cos\dfrac{7x}{2}-cos\dfrac{11x}{2}\right)\)
                  \(=2.sin\dfrac{21x}{2}.-2.sin\dfrac{9x}{2}.sin\left(-x\right)\)
                  \(=-4.sin\dfrac{21x}{2}.sin\dfrac{9x}{2}.-sinx=VP\left(dpcm\right)\)