A) \[\sin \theta \]
B) \[-\sin \theta \]
C) \[{{e}^{\cos \theta }}\]
D) \[{{e}^{\sin \theta }}\]
Correct Answer: B
Solution :
Let \[z={{e}^{{{e}^{-i\theta }}}}={{e}^{\cos \theta -i\sin \theta }}\]\[={{e}^{\cos \theta }}{{e}^{-i\sin \theta }}\] \[z={{e}^{\cos \theta }}[\cos (\sin \theta )-i\sin (\sin \theta )]\] \[z={{e}^{\cos \theta }}\cos (\sin \theta )-i{{e}^{\cos \theta }}\sin (\sin \theta )\] \[amp(z)={{\tan }^{-1}}\left[ -\frac{{{e}^{\cos \theta }}\sin (\sin \theta )}{{{e}^{\cos \theta }}\cos (\sin \theta )} \right]\] \[={{\tan }^{-1}}[\tan (-\sin \theta )]=-\sin \theta \].
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