Note that the double angle identity states that
[tex] \cos(2\alpha) = \cos^2(\alpha)-\sin^2(\alpha) [/tex]
Knowing the value of [tex] \cos(\alpha) [/tex], we can easily deduce the value of [tex] \sin^2(\alpha) [/tex] from the fundamental trigonometry identity:
[tex] \sin^2(\alpha)+\cos^2(\alpha)=1 \iff \sin^2(\alpha) = 1-\cos^2(\alpha)[/tex]
Which in your case evaluates to
[tex] \sin^2(\alpha) = 1-\cos^2(\alpha) = 1-\dfrac{25}{169} = \dfrac{144}{169}[/tex]
Now we can pulug these values in the double angle identity:
[tex] \cos(2\alpha) = \cos^2(\alpha)-\sin^2(\alpha) = \dfrac{25}{169} - \dfrac{144}{169} = -\dfrac{119}{169}[/tex]
