Il faut utiliser les formules d'Euler :
sin(5x)=Im [cos(5x)+isin(5x)]
= Im[e^(5ix)]
=Im[(e^(ix))^5]
=Im((cos(x)+isin(x))^5]
or(a+b)^5=a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+b^5
donc on obtient :
(cos(x)+isin(x))^5=(cos x)^5+5(cos x)^4(i sin x)+10(cos x)^3(i sin x)^2+
+10(cos x)^2(i sin x)^3+5(cos x)(i sin x)^4+(i sin x)^5
=(cos x)^5+5(cos x)^4(sin x) i -10(cos x)^3(sin x)^2+
-10 i (cos x)^2(sin x)^3+5(cos x)(sin x)^4-i (sin x)^5
donc sin(5x)=5(cos x)^4(sin x) -10 (cos x)^2(sin x)^3-(sin x)^5
