The period of f(x) = sin left( frac{pi x}{n - | vedclass

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(C) The given function is $f(x) = \sin \left( \frac{\pi x}{n - 1} \right) + \cos \left( \frac{\pi x}{n} \right)$.The period of $\sin \left( \frac{\pi

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$2n(n - 1)$

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(C) The given function is $f(x) = \sin \left( \frac{\pi x}{n - 1} \right) + \cos \left( \frac{\pi x}{n} \right)$.The period of $\sin \left( \frac{\pi x}{n - 1} \right)$ is $T_1 = \frac{2\pi}{\left( \frac{\pi}{n - 1} \right)} = 2(n - 1)$.The period of $\cos \left( \frac{\pi x}{n} \right)$ is $T_2 = \frac{2\pi}{\left( \frac{\pi}{n} \right)} = 2n$.The period of $f(x)$ is the $LCM$ of $T_1$ and $T_2$,which is the $LCM$ of $2(n - 1)$ and $2n$.Since $n$ and $n-1$ are coprime,the $LCM(2(n-1), 2n) = 2n(n - 1)$.Thus,the period is $2n(n - 1)$.

The period of $f(x) = \sin \left( \frac{\pi x}{n - 1} \right) + \cos \left( \frac{\pi x}{n} \right)$,where $n \in \mathbb{Z}$ and $n > 2$,is:

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