Question

evaluate 4sinx•sin(x+60°)•sin(x+120°)

Answer 1

Dear Student,

$$\begin{gathered} 4\sin \mathrm{x}.\sin \left(\mathrm{x}+60\right).\sin \left(\mathrm{x}+120\right) \\ =4\sin \mathrm{x}\left[\sin \mathrm{x}.\cos 60+\cos \mathrm{x}.\sin 60\right].\left[\sin \mathrm{x}.\cos 120+\cos \mathrm{x}.\sin 120\right] \\ =4\sin \mathrm{x}\left[\sin \mathrm{x}.\frac{1}{2}+\cos \mathrm{x}.\frac{\sqrt{3}}{2}\right]\left[\sin \mathrm{x}.-\frac{1}{2}+\cos \mathrm{x}.\frac{\sqrt{3}}{2}\right] \\ =4\sin \mathrm{x}\left[\frac{\sqrt{3}}{2}\cos \mathrm{x}+\frac{\sin \mathrm{x}}{2}\right]\left[\frac{\sqrt{3}}{2}\cos \mathrm{x}-\frac{\sin \mathrm{x}}{2}\right] \\ =4\sin \mathrm{x}\left[\frac{3}{4}{\cos }^2\mathrm{x}-\frac{{\sin }^2\mathrm{x}}{4}\right] \\ =4\sin \mathrm{x}\left[\frac{3}{4}\left(1-{\sin }^2\mathrm{x}\right)-\frac{{\sin }^2\mathrm{x}}{4}\right] \\ =4\sin \mathrm{x}\left[\frac{3}{4}-\frac{3{\sin }^2\mathrm{x}}{4}-\frac{{\sin }^2\mathrm{x}}{4}\right] \\ =4\sin \mathrm{x}\left[\frac{3}{4}-{\sin }^2\mathrm{x}\right] \\ =3\sin \mathrm{x}-4{\sin }^3\mathrm{x} \\ =\sin 3\mathrm{x} \end{gathered}$$

Regards