prove $\cos{ax}=\operatorname{F}(\frac{a}{2},\frac{-a}{2},\frac{1}{2};\sin^{2}{x})$

Question

How do I prove that : $$\cos{ax}=\operatorname{F}\left(\frac{a}{2},\frac{-a}{2},\frac{1}{2};\sin^{2}{x}\right)$$ I know that: $$\cos{ax}=\operatorname{F}\left(a,-a,\frac{1}{2};\frac{1-\cos{x}}{2}\right)\tag{1}$$ Thank you.

Answer 1

well, you know that: $$\cos2x=\cos^2x-\sin^2x=1-2\sin^2x$$ $$\Rightarrow \sin^2x=\frac{1-\cos 2x}{2}$$ now use the definition of the hypergeometric function and see what happens if you make the substitution $u=2x$