Answer
$$\cos2\theta=\frac{119}{169}$$
$$\sin2\theta=-\frac{120}{169}$$
Work Step by Step
$$\cos\theta=-\frac{12}{13}\hspace{2cm}\sin\theta\gt0$$
$$\sin2\theta=?\hspace{2cm}\cos2\theta=?$$
To find $\sin2\theta$ and $\cos2\theta$, it would be wise to find the unknown $\sin\theta$.
Using Pythagorean Identities:
$$\sin^2\theta=1-\cos^2\theta=1-\Big(-\frac{12}{13}\Big)^2=1-\frac{144}{169}=\frac{25}{169}$$
$$\sin\theta=\frac{5}{13}\hspace{1.5cm}(\sin\theta\gt0)$$
Now we apply the Double-Angle Identities for $\cos2\theta$ and $\sin2\theta$:
$$\cos2\theta=\cos^2\theta-\sin^2\theta=\Big(-\frac{12}{13}\Big)^2-\Big(\frac{5}{13}\Big)^2=\frac{144}{169}-\frac{25}{169}$$
$$\cos2\theta=\frac{119}{169}$$
$$\sin2\theta=2\sin\theta\cos\theta=2\times\frac{5}{13}\times\Big(-\frac{12}{13}\Big)$$
$$\sin2\theta=-\frac{120}{169}$$
