Let $A$ and $B$ denote the statements $\begin{array}{l} A:\cos \,\,\alpha + \cos \,\,\beta + \cos \,\,\gamma = 0\\ B:\sin \,\,\alpha + \sin \,\,\beta + \sin \,\,\gamma = 0 \end{array}$ If $\cos \,\,\left( {\beta - \gamma } \right) + \cos \left( {\gamma - \alpha } \right) + \cos \left( {\alpha - \beta } \right) = - \frac{3}{2}$, then

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Let

A

and

B

denote the statements

\begin{array}{l} A:\cos \,\,\alpha + \cos \,\,\beta + \cos \,\,\gamma = 0\\ B:\sin \,\,\alpha + \sin \,\,\beta + \sin \,\,\gamma = 0 \end{array}

\cos \,\,\left( {\beta - \gamma } \right) + \cos \left( {\gamma - \alpha } \right) + \cos \left( {\alpha - \beta } \right) = - \frac{3}{2}

, then

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Correct option: C

Both

A

and

B

are true

Option C (Both

A

and

B

are true) is correct because the given condition implies that both

(\cos \,\alpha + \cos \,\beta + \cos \,\gamma)^2

and

(\sin \,\alpha + \sin \,\beta + \sin \,\gamma)^2

must be zero, hence both statements are true.