If n in N and the period of frac{cos nx}{sin | vedclass

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(C) The period of a function $f(x) = \frac{g(x)}{h(x)}$ is the least common multiple of the periods of $g(x)$ and $h(x)$.Given $g(x) = \cos(nx)$,its p

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$2$

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(C) The period of a function $f(x) = \frac{g(x)}{h(x)}$ is the least common multiple of the periods of $g(x)$ and $h(x)$.Given $g(x) = \cos(nx)$,its period is $T_1 = \frac{2\pi}{n}$.Given $h(x) = \sin\left(\frac{x}{n}\right)$,its period is $T_2 = \frac{2\pi}{1/n} = 2n\pi$.The period of the quotient is the $LCM$ of $T_1$ and $T_2$.Since $T_2$ is a multiple of $T_1$ (as $2n\pi = n^2 \cdot \frac{2\pi}{n}$),the period of the function is $T_2 = 2n\pi$.Given $2n\pi = 4\pi$,we get $n = 2$.

If $n \in N$ and the period of $\frac{\cos nx}{\sin \left(\frac{x}{n}\right)}$ is $4\pi$,then $n$ is equal to

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