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FYI: I asked this question here couple of days ago but got no answer yet.

We know the global maximum of the function $\sin(nx)/\sin(x)$ is $n$ (thanks to this question), but what are the local maxima and minima points of the function in $(-\pi,\pi)$? (it represents the intensity of a narrow slit diffraction grating)

I tried equating the derivative to $0$ and the equation to be solved is $n\tan(x) = \tan(nx)$. I am unable to progress from here. I also tried expanding $\sin(nx)$ using multiple-angle formula but made no progress.

For convenience: 1st derivative $$ = \frac{n\cos(nx)\sin(x) - \sin(nx)\cos(x)}{\sin(x)^2}$$and 2nd derivative = $$\frac{2\sin(nx)\cos(x)^2 + \sin(nx)\sin(x)^2 - n^2\sin(nx)\sin(x)^2 - 2n\cos(nx)\cos(x)\sin(x)}{\sin(x)^3}$$

Here is a plot of the function in Desmos: https://www.desmos.com/calculator/kt0hntsbcb

FYI: I asked this question here couple of days ago but got no answer yet.

$n$ is an integer

We know the global maximum of the function $\sin(nx)/\sin(x)$ is $n$ (thanks to this question), but what are the local maxima and minima points of the function in $(-\pi,\pi)$? (it represents the intensity of a narrow slit diffraction grating)

I tried equating the derivative to $0$ and the equation to be solved is $n\tan(x) = \tan(nx)$. I am unable to progress from here. I also tried expanding $\sin(nx)$ using multiple-angle formula but made no progress.

For convenience: 1st derivative $$ = \frac{n\cos(nx)\sin(x) - \sin(nx)\cos(x)}{\sin(x)^2}$$and 2nd derivative = $$\frac{2\sin(nx)\cos(x)^2 + \sin(nx)\sin(x)^2 - n^2\sin(nx)\sin(x)^2 - 2n\cos(nx)\cos(x)\sin(x)}{\sin(x)^3}$$

Here is a plot of the function in Desmos: https://www.desmos.com/calculator/kt0hntsbcb

FYI: I asked this question here couple of days ago but got no answer yet.

We know the global maxima of the function $\sin(nx)/\sin(x)$ is $n$ (thanks to this question), but what are the local maxima and minima points of the function in ($-\pi,\pi$)? (it represents the intensity of a narrow slit diffraction grating)

I tried equating the derivative to $0$ and the equation to be solved is $n\tan(x) = \tan(nx)$. I am unable to progress from here. I also tried expanding $\sin(nx)$ using multiple-angle formula but made no progress.

For convenience: 1st derivative $$ = \frac{n\cos(nx)\sin(x) - \sin(nx)\cos(x)}{\sin(x)^2}$$and 2nd derivative = $$\frac{2\sin(nx)\cos(x)^2 + \sin(nx)\sin(x)^2 - n^2\sin(nx)\sin(x)^2 - 2n\cos(nx)\cos(x)\sin(x)}{\sin(x)^3}$$

Here is a plot of the function in Desmos: https://www.desmos.com/calculator/kt0hntsbcb

FYI: I asked this question here couple of days ago but got no answer yet.

We know the global maximum of the function $\sin(nx)/\sin(x)$ is $n$ (thanks to this question), but what are the local maxima and minima points of the function in $(-\pi,\pi)$? (it represents the intensity of a narrow slit diffraction grating)

I tried equating the derivative to $0$ and the equation to be solved is $n\tan(x) = \tan(nx)$. I am unable to progress from here. I also tried expanding $\sin(nx)$ using multiple-angle formula but made no progress.

For convenience: 1st derivative $$ = \frac{n\cos(nx)\sin(x) - \sin(nx)\cos(x)}{\sin(x)^2}$$and 2nd derivative = $$\frac{2\sin(nx)\cos(x)^2 + \sin(nx)\sin(x)^2 - n^2\sin(nx)\sin(x)^2 - 2n\cos(nx)\cos(x)\sin(x)}{\sin(x)^3}$$

Here is a plot of the function in Desmos: https://www.desmos.com/calculator/kt0hntsbcb