<s>[INST] What property was used in this sine transformation? I have this expression:
$$
ψ(χ) = A\sin^3(\frac{πχ}{α})
$$
And somehow the book i read equalizes the previous equation to this one:
$$
ψ(χ) = \frac{A}{4}[3\sin(\frac{πχ}{α}) - \sin(\frac{3πχ}{α})]
$$
What trigonometric identity was used to make this possible?
 [/INST] By DeMoivre's identity, 
$\cos(3x)+i\sin(3x)=(\cos(x)+i\sin(x))^3=$
$\cos^3(x)+3i\cos^2(x)\sin(x)-3\cos(x)\sin^2(x)-i\sin^3(x)$.
Grouping the imaginary terms together, we see that $\sin(3x)=3\cos^2(x)\sin(x)-\sin^3(x)$.
We can rewrite $3\cos^2(x)\sin(x)$ as $3\sin(x)(1-\sin^2(x))$, which gives us $\sin(3x)=3\sin(x)-4\sin^3(x)$.
Finally, we can write $\sin^3(x)=\frac{1}{4}(3\sin(x)-\sin(3x))$.
</s>