Is the following inequality true? $$\left|\sum_{i=1}^{n}\left(\cos(x_i) \prod_{j\neq i}\sin(x_j)\right)\right|\le 1$$

I tried to count the extremes but it didn't work.


This is not true in general, for example:

take $x=y=z = \dfrac{\pi}{3}$, we can see

$$\cos(x)\sin(y)\sin(z) + \cos(y)\sin(x)\sin(z) + \cos(z)\sin(x)\sin(y) = 3 \dfrac{1}{2}\dfrac{\sqrt{3}}{2}\dfrac{\sqrt{3}}{2} =\dfrac{9}{8} > 1$$


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