TITLE: How to prove $4\times{_2F_1}(-1/4,3/4;7/4;(2-\sqrt3)/4)-{_2F_1}(3/4,3/4;7/4;(2-\sqrt3)/4)\stackrel?=\frac{3\sqrt[4]{2+\sqrt3}}{\sqrt2}$
QUESTION [22 upvotes]: I have the following conjecture, which is supported by numerical calculations up to at least $10^5$ decimal digits:
$$4\times{_2F_1}\left(-\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{2-\sqrt{3}}{4}\right)-{_2F_1}\left(\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{2-\sqrt{3}}{4}\right)\,\stackrel?=\,\frac{3\sqrt[4]{2+\sqrt{3}}}{\sqrt{2}},$$
where $_2F_1$ denotes the hypergeometric function.
Can you suggest any ideas how to prove it?

The conjectural closed form was obtained using WolframAlpha query
ToRadicals[RootApproximant[2.94844576626425580599908814238570067699233]]

REPLY [21 votes]: This is the identity 15.5.12 from DLMF, with $a=-1/4$, $b=3/4$, $c=7/4$ and the special form
$$ F(b,a,a,x)=(1-x)^{-b}. $$
Is this how you got your identity?