Question

$\left|\begin{array}{ccc}{\sin }^{3}A& {\sin }^{3}B& {\sin }^{3}C\\ \sin A& \sin B& \sin C\\ \cos A& \cos B& \cos C\end{array}\right|$ $=-\sin (A-B)\sin (B-C)\sin (C-A)\sin (A+B+C)$ where $A,B,C$ are any three angles such that $A+B+C$ $\ne n\pi$

• A. True
• B. False

Answer 1

The correct option is A True

${\sin }^{3}A\left[\sin \right(B-C\left)\right]-{\sin }^{3}B\left[\sin \right(A-C\left)\right]+{\sin }^{3}C\left[\sin \right(A-B\left)\right]$

$=-\sin (A-B)\sin (B-C)\sin (C-A)\sin (A+B+C)$

$\Rightarrow {\sin }^{3}C.\sin (A-B)+{\sin }^{3}A.\sin (B-C)+{\sin }^{3}B\sin (C-A)$

$=RHS$