You can easily notice that $4\leq n\leq7$ and then square first equation obtain $sinx$ in terms of $cosx$ then put $sinx$ in same equation, solve square equation, obtain $cosx$ in dependence of $n$. Notice that $tg\frac{x}{2}=\sqrt{\frac{1-cosx}{1+cosx}}.$ Thus we can put $cosx$ in second equation and obtain an equation for $n.$ Now we can easily check that for $4\leq n\leq7$ only $4$ and $7$ satisfy the equation. Now we can easily get solutions $(x,n)=(0,4);\left(\frac{arcsin\frac{3}{4}}{2},7\right)$

Quick question, what is tg?

TheDonkTonk wrote: Quick question, what is tg? $\tan$

It was an amazing TST, I have got so much satisfaction after it (no)

kirillnaval wrote: Let $n$ be a natural number. Find all solutions $x$ of the system of equations $$\left\{\begin{matrix} sinx+cosx=\frac{\sqrt{n}}{2}\\tg\frac{x}{2}=\frac{\sqrt{n}-2}{3}\end{matrix}\right.$$On interval $\left[0,\frac{\pi}{4}\right).$

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