Question

[Capheny Bản Quyền]

\(sin\dfrac{x}{2}sinx-cos\dfrac{x}{2}sin^2x+1=2cos^2\left(\dfrac{pi}{4}-\dfrac{x}{2}\right)\)

Answer 1

[Nguyễn Việt Lâm]

\(sin\dfrac{x}{2}sinx-cos\dfrac{x}{2}sin^2x=2cos^2\left(\dfrac{\pi}{4}-\dfrac{x}{2}\right)-1\)

\(\Leftrightarrow sin\dfrac{x}{2}sinx-cos\dfrac{x}{2}sin^2x=cos\left(\dfrac{\pi}{2}-x\right)\)

\(\Leftrightarrow sin\dfrac{x}{2}sinx-cos\dfrac{x}{2}sin^2x=sinx\)

\(\Leftrightarrow\left[{}\begin{matrix}sinx=0\Rightarrow x=k\pi\\sin\dfrac{x}{2}-cos\dfrac{x}{2}.sinx=1\left(1\right)\end{matrix}\right.\)

Xét (1)

\(\Leftrightarrow sin\dfrac{x}{2}-2sin\dfrac{x}{2}.cos^2\dfrac{x}{2}=1\)

\(\Leftrightarrow sin\dfrac{x}{2}-2sin\dfrac{x}{2}\left(1-sin^2\dfrac{x}{2}\right)=1\)

\(\Leftrightarrow2sin^3\dfrac{x}{2}-sin\dfrac{x}{2}-1=0\)

\(\Leftrightarrow\left(sin\dfrac{x}{2}-1\right)\left(2sin^2\dfrac{x}{2}+2sin\dfrac{x}{2}+1\right)=0\)

\(\Leftrightarrow sin\dfrac{x}{2}=1\Leftrightarrow...\)

Answer 2

[Julian Edward]