#### Answer

$\displaystyle \cos 2x=-\frac{8}{17}$
$\displaystyle \sin 2x=\frac{15}{17}$

#### Work Step by Step

Plan: work out sinx and cosx, and then apply the double-angle formulas.
$\tan x$ is positive and $\sin x$ is negative $\Rightarrow \cos x$ is negative,
since $\displaystyle \tan x=\frac{\sin x}{\cos x}$.
This also means that x is in quadrant III,
where both its sine and cosine are negative.
Pythagorean Identity:
$\displaystyle \sec^{2}x=1+\tan^{2}x=1+(\frac{5}{3})^{2}=1+\frac{25}{9}=\frac{34}{9}$
... cosine and secant are reciprocals, so
$\displaystyle \cos^{2}x=\frac{9}{34}$ and, since $\cos x$ is negative,
$\displaystyle \cos x=-\frac{3}{\sqrt{34}}\cdot\frac{\sqrt{34}}{\sqrt{34}}=-\frac{3\sqrt{34}}{34}$
Pythagorean Identity ($\sin x$ is negative):
$\sin x=-\sqrt{1-\cos^{2}x}$
$=-\sqrt{1-(-\dfrac{3}{\sqrt{34}})^{2}}$
$=-\displaystyle \sqrt{1-\frac{9}{34}}=-\sqrt{\frac{25}{34}}=-\frac{5}{\sqrt{34}}\cdot\frac{\sqrt{34}}{\sqrt{34}}$
$=-\dfrac{5\sqrt{34}}{34}$
Finally, the Double-Angle Identities:
$\displaystyle \cos 2x=2\cos^{2}x-1=2(\frac{9}{34})-1=\frac{9}{17}-1=-\frac{8}{17}$
$\displaystyle \sin 2x=2\sin x\cos x=2(-\frac{5\sqrt{34}}{34})(-\frac{3\sqrt{34}}{34})=\frac{15}{17}$