A) \[\cos \,(\pi /4)+i\,sin\,(\pi /4)\]
B) \[\cos \,(\pi /2)+i\,\sin \,(\pi /2)\]
C) \[\sqrt{2}\,(\cos \,(\pi /4)+i\,\sin (\pi /4))\]
D) \[\sqrt{2}\,(\cos \,(\pi /2)\,+\,i\,\sin (\pi /2))\]
Correct Answer: D
Solution :
\[\frac{i+1}{\cos \frac{\pi }{4}-i\sin \frac{\pi }{4}}=\frac{i+1}{\frac{1}{\sqrt{2}(1-i)}}\times \frac{1+i}{1+i}\] \[=\frac{\sqrt{2}{{(1+i)}^{2}}}{1-{{i}^{2}}}\] \[=\frac{\sqrt{2}(1-1+2i)}{1+1}\] \[=\frac{\sqrt{2}.2i}{2}=\sqrt{2}i\] \[=\sqrt{2}\left( \cos \,\frac{\pi }{2}+i\,\sin \frac{\pi }{2} \right)\]
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