If α+ β=0= α+ β , α≠β then what is a value of 2α+ 2β+2 (α+β) ?

Let $z_1 = \cos\alpha + i\sin\alpha$ and $z_2 = \cos\beta + i\sin\beta$.

From the given conditions:

$z_1 + z_2 = (\cos\alpha + \cos\beta) + i(\sin\alpha + \sin\beta) = 0$

Squaring both sides:

$(z_1 + z_2)^2 = 0$

$z_1^2 + z_2^2 + 2z_1 z_2 = 0$

Using De Moivre's theorem, $z_1^2 = \cos 2\alpha + i\sin 2\alpha$ and $z_2^2 = \cos 2\beta + i\sin 2\beta$. The product term is $2z_1 z_2 = 2(\cos(\alpha+\beta) + i\sin(\alpha+\beta))$.

Substituting these into the squared equation:

$(\cos 2\alpha + \cos 2\beta + 2\cos(\alpha+\beta)) + i(\sin 2\alpha + \sin 2\beta + 2\sin(\alpha+\beta)) = 0$

Equating the real part to zero:

$\cos 2\alpha + \cos 2\beta + 2\cos(\alpha+\beta) = 0$

Answer: $0$