If \( f(x)=\sin x+\tan \frac{x}{2}+\sin \frac{x}{2^{2}}+\tan \frac{x}{2^{3}}+\ldots+\sin \frac{x...
If \( f(x)=\sin x+\tan \frac{x}{2}+\sin \frac{x}{2^{2}}+\tan \frac{x}{2^{3}}+\ldots+\sin \frac{x}{2^{n-1}} \) \( +\tan \frac{x}{2^{n}} \) is a periodic function with period \( k \pi \), then \( k= \)
(A) 1
(B) 2
(C) \( 2^{n} \)
(D) \( \frac{1}{2^{n}} \)
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