Question

chứng minh \(sin^3x+cos^3x=\frac{3\left(sinx+cosx\right)-\left(sinx+cos\right)^3}{2}\)

Answer 1

Lời giải:
\(3(\sin x+\cos x)-(\sin x+\cos x)^3=(\sin x+\cos x)[3-(\sin x+\cos x)^2]\)

\(=(\sin x+\cos x)[3-(\sin ^2x+\cos ^2x)-2\sin x\cos x]\)

\(=(\sin x+\cos x)(3-1-2\sin x\cos x)=2(\sin x+\cos x)(1-\sin x\cos x)=2(\sin x+\cos x)(\sin ^2x+\cos ^2x-\sin x\cos x)\)

\(=2(\sin ^3+\cos ^3x)\)

\(\Rightarrow \frac{3(\sin x+\cos x)-(\sin x+\cos x)^3}{2}=\sin ^3x+\cos ^3x\)(đpcm)