A) \[i\]
B) \[-i\]
C) \[\frac{1+\sqrt{3}i}{2}\]
D) \[\frac{1-\sqrt{3}i}{2}\]
E) \[1+i\]
Correct Answer: A
Solution :
\[LHS=\frac{\cos 30{}^\circ +i\sin 30{}^\circ }{\cos 60{}^\circ -i\sin 60{}^\circ }\] \[=\frac{\frac{\sqrt{3}}{2}+i\frac{1}{2}}{\frac{1}{2}-i\frac{\sqrt{3}}{2}}=\frac{\frac{\sqrt{3}+i}{2}}{\frac{1-i\sqrt{3}}{2}}\] \[=\frac{\sqrt{3}+i}{1-i\sqrt{3}}\times \frac{1+i\sqrt{3}}{1+i\sqrt{3}}=\frac{\sqrt{3}+3i+i-\sqrt{3}}{1+3}\] \[=\frac{4i}{4}=i\]
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