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?
 A: 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
$$
\frac{\Gamma\left(\frac2{25}\right)\Gamma\left(\frac{7}{25}\right)\Gamma\left(\frac{12}{25}\right)}{\Gamma\left(\frac25\right)\Gamma\left(\frac3{25}\right)\Gamma\left(\frac8{25}\right)}=4\cdot5^{1/10}\sin\left(\frac{3\pi}{25}\right)\sin\left(\frac{8\pi}{25}\right)
$$
A: Using Maple, I get
$$
{\frac {{5}^{1/10} \left( 2\,\cos \left( {\frac {8\,\pi }{25}}
 \right) +2\,\cos \left( {\frac {4\,\pi }{25}} \right) +1 \right) }{
2\;\cos \left( {\frac {6\,\pi }{25}} \right) +2\;\cos \left( \frac{\pi}{5} \right) 
}}
$$
