Question Number 75504 by mathocean1 last updated on 12/Dec/19
$$\mathrm{please}\:\mathrm{sirs}\:\mathrm{i}\:\mathrm{would}\:\mathrm{like}\:\mathrm{that}\:\mathrm{you}\:\mathrm{help}\:\mathrm{me}\:\mathrm{to} \\ $$$$\mathrm{show}\:\mathrm{this}: \\ $$$$\mathrm{sin}\left(\frac{\pi}{\mathrm{3}}+\mathrm{x}\right)\mathrm{sin}\left(\frac{\pi}{\mathrm{3}\:}−\mathrm{x}\right)=\frac{\mathrm{3}}{\mathrm{4}}−\mathrm{sin}^{\mathrm{2}} \mathrm{x} \\ $$
Answered by som(math1967) last updated on 12/Dec/19
$${sin}\left(\frac{\pi}{\mathrm{3}}\:+{x}\right){sin}\left(\frac{\pi}{\mathrm{3}}−{x}\right) \\ $$$$={sin}^{\mathrm{2}} \frac{\pi}{\mathrm{3}}\:−{sin}^{\mathrm{2}} {x}\:\bigstar \\ $$$$=\frac{\mathrm{3}}{\mathrm{4}}\:−{sin}^{\mathrm{2}} {x} \\ $$$$\bigstar{sin}\left({A}+{B}\right){sin}\left({A}−{B}\right)={sin}^{\mathrm{2}} {A}−{sin}^{\mathrm{2}} {B} \\ $$
Commented by mathocean1 last updated on 12/Dec/19
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{sir} \\ $$