A) \[0\]
B) \[\cos (\alpha +\beta +\gamma )\]
C) \[3\cos (\alpha +\beta +\gamma )\]
D) \[3\sin (\alpha +\beta +\gamma )\]
Correct Answer: C
Solution :
Let\[a=\cos \alpha +i\sin \alpha \], \[b=\cos \beta +i\sin \beta \], \[c=\cos \gamma +i\sin \gamma \] then;\[a+b+c=(\cos \alpha +\cos \beta +\cos \gamma )+\] \[i(\sin \alpha +\sin \beta +\sin \gamma )\] \[=0+i0=0\] \[\Rightarrow \]\[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}=3abc\] \[\Rightarrow \]\[(\cos 3\alpha +i\sin 3\alpha )+(\cos 3\beta +i\sin 3\beta )\] \[+(\cos 3\gamma +i\sin 3\gamma )\] \[=3[\cos (\alpha +\beta +\gamma )\,+i\sin (\alpha +\beta +\gamma )]\] \[\Rightarrow \,\,\cos 3\alpha +\cos 3\beta +\cos 3\gamma =3\cos (\alpha +\beta +\gamma )\]
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