If $\sin A+\sin B+\sin C=\cos A+\cos B+\cos C=0$, prove that:…

If $\sin A+\sin B+\sin C=\cos A+\cos B+\cos C=0$, prove that: $\cos 3A+\cos 3B+ \cos 3C=3\cos(A+B+C)$.

My Attempt;

Here,

$$e^{iA}=\cos A+i\sin A$$
$$e^{iB}=\cos B+i\sin B$$
$$e^{iC}=\cos C+i\sin C$$

Then,
$$e^{iA}+e^{iB}+e^{iC}=0$$

Now, what should I do further. Please help.

Solutions Collecting From Web of "If $\sin A+\sin B+\sin C=\cos A+\cos B+\cos C=0$, prove that:…"

Hint: Use $a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$ by taking $a=e^{iA},b=e^{iB}$ and $c=e^{iC}$.

let $a = e^{iA}, b = e^{iB}, c = e^{iC}$. Also note that $a+b+c = 0$
$$a^3 + b^3 + c^3 – 3abc = (a+b+c)(a^2 + b^2 + c^2 – ab – bc -ca) = 0$$
$$a^3 + b^3 + c^3 = 3abc$$
$$\mathrm{cis}\ 3A + \mathrm{cis}\ 3B + \mathrm{cis}\ 3C = 3\ \mathrm{cis}\ (a+b+c)$$
equating the real parts will give you the required expression:
$$\mathrm{cos}\ 3A + \mathrm{cos}\ 3B + \mathrm{cos}\ 3C = 3\ \mathrm{cos}\ (a+b+c)$$