Verified answer

[tex]\displaystyle\bf\\f(x)=Cos^{2} \frac{x}{8} -Sin^{2} \frac{x}{8} =Cos\Big(2\cdot \frac{x}{8} \Big)=Cos\frac{x}{4} \\\\\\F(x)=4Sin\frac{x}{4} +C[/tex]