Question

If $A,B,C,P,Q,RϵR$, then

$\Delta =\left|\begin{array}{ccc}\cos (A+P)& \cos (A+Q)& \cos (A+R)\\ \cos (B+P)& \cos (B+Q)& \cos (B+R)\\ \cos (C+P)& \cos (C+Q)& \cos (C+R)\end{array}\right|$

equals to

• A. $\cos (A+B+C)+\cos (P+Q+R)$
• B. $\cos (A+P)\cos (B+Q)\cos (C+R)$
• C. $\cos A\cos P+\cos B\cos Q+\cos C\cos R$
• D. none of these

Answer 1

The correct option is D none of these

$\Delta =\left|\begin{array}{ccc}\cos (A+P)& \cos (A+Q)& \cos (A+R)\\ \cos (B+P)& \cos (B+Q)& \cos (B+R)\\ \cos (C+P)& \cos (C+Q)& \cos (C+R)\end{array}\right|$

$=\left|\begin{array}{ccc}\cos A\cos P-\sin A\sin P& \cos A\cos Q-\sin A\sin Q& \cos A\cos R-\sin A\sin R\\ \cos B\cos P-\sin B\sin P& \cos B\cos Q-\sin B\sin Q& \cos B\cos R-\sin B\sin R\\ \cos C\cos P-\sin C\sin P& \cos C\cos Q-\sin C\sin Q& \cos C\cos R-\sin C\sin R\end{array}\right|$

[$\because \cos (A+B)=\cos A\cos B-\sin A\sin B$]

$=\left|\begin{array}{ccc}\cos A\cos P& \cos A\cos Q& \cos A\cos R\\ \cos B\cos P& \cos B\cos Q& \cos B\cos R\\ \cos C\cos P& \cos C\cos Q& \cos C\cos R\end{array}\right|-\left|\begin{array}{ccc}\sin A\sin P& \sin A\sin Q& \sin A\sin R\\ \sin B\sin P& \sin B\sin Q& \sin B\sin R\\ \sin C\sin P& \sin C\sin Q& \sin C\sin R\end{array}\right|$

$=\left(\cos P\cos Q\cos R\right)\left|\begin{array}{ccc}\cos A& \cos A& \cos A\\ \cos B& \cos B& \cos B\\ \cos C& \cos C& \cos C\end{array}\right|-\left(\sin P\sin Q\sin R\right)\left|\begin{array}{ccc}\sin A& \sin A& \sin A\\ \sin B& \sin B& \sin B\\ \sin C& \sin C& \sin C\end{array}\right|$

$=\left(\cos P\cos Q\cos R\right)\times 0-\left(\sin P\sin Q\sin R\right)\times 0$

$=0$