first, we need to compute the hypotenuse (c) of the triangle. Using the Pythagorean theorem:
[tex]\begin{gathered} c^2=a^2+b^2 \\ c^2=5^2+12^2 \\ c^2=25+144 \\ c=\sqrt[]{169} \\ c=13 \end{gathered}[/tex]
By definition:
[tex]\cos (angle)=\frac{\text{adjacent side}}{hypotenuse}[/tex]
Replacing with data:
[tex]\cos (\alpha)=\frac{5}{13}[/tex]
Using the next identity:
[tex]\begin{gathered} \cos (2\alpha)=2\cos ^2(\alpha)-1 \\ \cos (2\alpha)=2(\frac{5}{13})^2-1 \\ \cos (2\alpha)=\frac{50}{169}^{}-1 \\ \cos (2\alpha)=-\frac{119}{169} \end{gathered}[/tex]
