# Trigonometric equations

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# The positive integer value of n > 3 satisfying the equation $\frac{1}{\mathrm{sin}\left(\frac{\mathrm{\pi }}{\mathrm{n}}\right)}=\frac{1}{\mathrm{sin}\left(\frac{2\mathrm{\pi }}{\mathrm{n}}\right)}+\frac{1}{\mathrm{sin}\left(\frac{3\mathrm{\pi }}{\mathrm{n}}\right)}$ is ________.

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Solution

## $\frac{1}{\mathrm{sin}\frac{\mathrm{\pi }}{\mathrm{n}}}-\frac{1}{\mathrm{sin}\frac{3\mathrm{\pi }}{\mathrm{n}}}=\frac{1}{\mathrm{sin}\frac{2\mathrm{\pi }}{\mathrm{n}}}$or   $\frac{\mathrm{sin}\frac{3\mathrm{\pi }}{\mathrm{n}}-\mathrm{sin}\frac{\mathrm{\pi }}{\mathrm{n}}}{\mathrm{sin}\frac{\mathrm{\pi }}{\mathrm{n}}\mathrm{sin}\frac{3\mathrm{\pi }}{\mathrm{n}}}=\frac{1}{\mathrm{sin}\frac{2\mathrm{\pi }}{\mathrm{n}}}$or   $\frac{\left(2\mathrm{sin}\frac{\mathrm{\pi }}{\mathrm{n}}\mathrm{cos}\frac{2\mathrm{\pi }}{\mathrm{n}}\right)\mathrm{sin}\frac{2\mathrm{\pi }}{\mathrm{n}}}{\mathrm{sin}\frac{\mathrm{\pi }}{\mathrm{n}}\mathrm{sin}\frac{3\mathrm{\pi }}{\mathrm{n}}}=1$or   $\mathrm{sin}\frac{4\mathrm{\pi }}{\mathrm{n}}=\mathrm{sin}\frac{3\mathrm{\pi }}{\mathrm{n}}$

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Number of solutions of the equation $4\left({\mathrm{cos}}^{2}2\mathrm{x}+\mathrm{cos}2\mathrm{x}+1\right)+\mathrm{tan}\mathrm{x}\left(\mathrm{tan}\mathrm{x}-2\sqrt{3}\right)=0$ in  is