\(cos(2A)=2cos^2(A)-1\)
Note \(cos^2(A)=1-sin^2(A)\) (Pythagoras identity, \(sin^2(x)+cos^2(x)=1\) rearrange and solve for \(cos^2(x)\))
\(2(1-sin^2(A))-1\)
\(cos(2A)=2-2sin^2(A)-1\)
Given \(sin(A)=\frac{12}{13}\) , then \(sin^2(A)=\frac{144}{169}\)
Substitute
\(cos(2A)=2-2(\frac{144}{169})-1\)
\(cos(2A)=-\frac{119}{169}=-0.704142\)