Question

cho phương trình \(\dfrac{1}{sinx}+\dfrac{1}{sin2x}+\dfrac{1}{sin4x}+...+\dfrac{1}{sin2^{2018}x}=0\)

Answer 1

\(\dfrac{1}{sin2k}=\dfrac{sink}{sink.sin2k}=\dfrac{\left(sin2k-k\right)}{sink.sin2k}=\dfrac{sin2k.cosk-cos2k.sink}{sink.sin2k}\)

\(=\dfrac{cosk}{sink}-\dfrac{cos2k}{sin2k}=cotk-cot2k\)

Do đó pt tương đương:

\(cot\dfrac{x}{2}-cotx+cotx-cot2x+...+cot2^{2017}x-cot^{2018}x=0\)

\(\Leftrightarrow cot\dfrac{x}{2}-cot2^{2018}x=0\)

\(\Leftrightarrow\dfrac{x}{2}=2^{2018}x+k\pi\)

\(\Leftrightarrow...\)

Answer 2

@Nguyễn VIệt Lâm giúp em với