# Simplification of a combination of 6 values of the gamma function

I'm trying to simplify this combination of gamma functions: $$\frac{\Gamma\left(\frac{2}{25}\right)\Gamma\left(\frac{7}{25}\right)\Gamma\left(\frac{12}{25}\right)}{\Gamma\left(\frac{2}{5}\right)\Gamma\left(\frac{3}{25}\right)\Gamma\left(\frac{8}{25}\right)}$$ I tried to apply Gauss's multiplication formula, as it was done in this answer, but without any success. It is possible to simplify this expression at all?

Using the Gauss Multiplication Formula, which is proven in this answer, $$\Gamma\left(\frac2{25}\right)\Gamma\left(\frac{7}{25}\right)\Gamma\left(\frac{12}{25}\right)\Gamma\left(\frac{17}{25}\right)\Gamma\left(\frac{22}{25}\right)=4\pi^25^{1/10}\Gamma\left(\frac25\right)$$ Using the Euler Reflection Formula, proven at the end of this answer, $$\Gamma\left(\frac3{25}\right)\Gamma\left(\frac{22}{25}\right)=\frac\pi{\sin\left(\frac{3\pi}{25}\right)}$$ and $$\Gamma\left(\frac8{25}\right)\Gamma\left(\frac{17}{25}\right)=\frac\pi{\sin\left(\frac{8\pi}{25}\right)}$$ Putting these together, we get $$\frac{\Gamma\left(\frac2{25}\right)\Gamma\left(\frac{7}{25}\right)\Gamma\left(\frac{12}{25}\right)}{\Gamma\left(\frac25\right)\Gamma\left(\frac3{25}\right)\Gamma\left(\frac8{25}\right)}=4\cdot5^{1/10}\sin\left(\frac{3\pi}{25}\right)\sin\left(\frac{8\pi}{25}\right)$$
$${\frac {{5}^{1/10} \left( 2\,\cos \left( {\frac {8\,\pi }{25}} \right) +2\,\cos \left( {\frac {4\,\pi }{25}} \right) +1 \right) }{ 2\;\cos \left( {\frac {6\,\pi }{25}} \right) +2\;\cos \left( \frac{\pi}{5} \right) }}$$