A) \[\cos (4\alpha +5\beta )+i\,\sin (4\alpha +5\beta )\]
B) \[\cos (4\alpha +5\beta )-i\,\sin (4\alpha +5\beta )\]
C) \[\sin (4\alpha +5\beta )-i\cos (4\alpha +5\beta )\]
D) None of these
Correct Answer: C
Solution :
\[\frac{{{(\cos \alpha +i\sin \alpha )}^{4}}}{{{(\sin \beta +i\cos \beta )}^{5}}}\] \[=\frac{\cos 4\alpha +i\sin 4\alpha }{{{i}^{5}}{{(\cos \beta -i\sin \beta )}^{5}}}\] = \[-i\,(\cos 4\alpha +i\sin 4\alpha )\,{{(\cos \beta -i\sin \beta )}^{-5}}\] = \[-i\,[\cos 4\alpha +i\sin 4\alpha ]\,\,[\cos 5\beta +i\sin 5\beta ]\] = \[-i\,[\cos (4\alpha +5\beta )+i\sin (4\alpha +5\beta )]\] = \[\sin (4\alpha +5\beta )-i\,\cos (4\alpha +5\beta )\].
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