| Three expressions are given below: |
| \[{{Q}_{1}}=\sin (A+B)+\sin (B+C)+\sin (C+A)\] |
| \[{{Q}_{2}}=\cos (A-B)+\cos (B-C)+\cos (C-A)\] |
| \[{{Q}_{3}}=\sin A(\cos B+\cos C)+\sin B(\cos C+\cos A)+\]\[\sin C(\cos A+\cos B)\] |
| Which one of the following is correct? |
A) \[{{Q}_{1}}={{Q}_{2}}\]
B) \[{{Q}_{2}}={{Q}_{3}}\]
C) \[{{Q}_{1}}={{Q}_{3}}\]
D) All the expressions are different
Correct Answer: C
Solution :
We take \[{{Q}_{3}}\]first, \[{{Q}_{3}}=\sin A(\cos B+\cos C)+\sin B(\cos C+\cos A)\]\[+\sin C(\cos A+\cos B)\] \[=\sin A\operatorname{cosB}+sinAcosC+sinBcosC+sinBcosA\]\[+\sin C\cos A+\sin C\cos B\] \[=\sin (A+B)+\sin (B+C)+\sin (C+A)={{Q}_{1}}\] \[\Rightarrow \,\,{{Q}_{3}}={{Q}_{1}}\]
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