If $α,β$ and $γ$ are such that $α+β+γ=0$, then $\begin{vmatrix}1&\cos γ&\cos β\\\cos γ&1&\cos α\\\cos β&\cos α&1\end{vmatrix}$ is equal to

none of these

Let A, B and C be three numbers such that $α = B-C, β=C - A$ and $γ = A - B$. Clearly, $α+β+γ=0$.

$∴\begin{vmatrix}1&\cos (A-B)&\cos (C-A)\\\cos (A-B)&1&\cos (B-C)\\\cos (C-A)&\cos (B-C)&1\end{vmatrix}$

$=\begin{vmatrix}\cos^2 A + \sin^2 A&\cos A \cos B+ \sin A \sin B &\cos A \cos C+ \sin A sin C\\\cos A \cos B+ \sin A \sin B&\cos^2 B + \sin^2 B&\cos B \cos C + \sin B \sin C\\\cos C \cos A+ \sin C \sin A&\cos B \cos C + \sin B \sin C&\cos^2 C + \sin^2 C\end{vmatrix}$

$=\begin{vmatrix}\cos A &\sin A&0\\\cos B &\sin B&0\\\cos C &\sin C&0\end{vmatrix}\begin{vmatrix}\cos A &\sin A&0\\\cos B &\sin B&0\\\cos C &\sin C&0\end{vmatrix}=0×0=0$