A) \[\sin u\,\,\cos v\,\,[\cos (x+y-u-v)+\]\[i\sin (x+y-u-v)]\]
B) \[\sin \,u\,\cos \,v[\cos \,(x+y+u+v)+\]\[i\sin (x+y+u+v)]\]
C) \[\sin \,\,u\,\,\,\cos \,\,v\,\,[\cos \,(x+y+u+v)-\]\[i\sin (x+y-u+v)]\]
D) None of these
Correct Answer: A
Solution :
Given \[\frac{(\cos x+i\sin x)(\cos y+i\sin y)}{(\cot u+i)(1+i\tan \,\,v)}\] \[=\,\,\frac{(cos\,x+i\,sin\,x)(cos\,y+i\,sin\,y)}{(cos\,u+i\,sin\,u)(cos\,\nu +i\,sin\,\nu )}\] \[= sin u cos \nu [cos \left( x +y - u - v \right)\] \[+\,\,i\,\sin \left( x+y-u-\nu \right)]\]
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