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 the above
Correct Answer: A
Solution :
\[LHS\] \[=\frac{(\cos x+i\sin x)(\cos y+i\sin y)}{(\cos u+i\sin u)(\cos v+i\sin v)}\sin u\cos v\] \[=\sin u\cos v\,\,{{e}^{ix}}\cdot {{e}^{iy}}\cdot {{e}^{-iu}}\cdot {{e}^{-iv}}\] \[=\sin u\cos v[\cos (x+y-u-v)\]\[+i\sin (x+y-u-v)]\]
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