Question

\(sin^3x+2sin^2x.cosx-3cos^3x=0\)

Answer 1

\(\Leftrightarrow sin^3x-sin^2x.cosx+3\left(sin^2x.cosx-cos^3x\right)=0\)

\(\Leftrightarrow sin^2x\left(sinx-cosx\right)+\left(sinx-cosx\right)\left(3sinx.cosx+3cos^2x\right)=0\)

\(\Leftrightarrow\left(sinx-cosx\right)\left(sin^2x+3sinx.cosx+3cos^2x\right)=0\)

\(\Leftrightarrow\sqrt{2}sin\left(x-\frac{\pi}{4}\right)\left[\left(sinx+\frac{3}{2}cosx\right)^2+\frac{3}{4}cos^2x\right]=0\)

\(\Leftrightarrow sin\left(x-\frac{\pi}{4}\right)=0\)

\(\Leftrightarrow x=\frac{\pi}{4}+k\pi\)