Answer
$\displaystyle \cos 2\theta=\frac{17}{25}$
$\sin 2\displaystyle \theta=-\frac{4\sqrt{21}}{25}$
Work Step by Step
First, using the Pythagorean Identity$ \sin^{2}\theta+\cos^{2}\theta=1,$
with $\cos\theta < 0,$
$\displaystyle \cos\theta=-\sqrt{1-(\frac{2}{5})^{2}}=-\sqrt{1-\frac{4}{25}}=-\sqrt{\frac{21}{25}}=-\frac{\sqrt{21}}{5}$
We now use the Double-Angle Identities:
$\displaystyle \cos 2\theta=\cos^{2}\theta-\sin^{2}\theta=\frac{21}{25}-\frac{4}{25}=\frac{17}{25}$
$\sin 2\displaystyle \theta=2\sin\theta\cos\theta=2\cdot\frac{-\sqrt{21}}{5}\cdot\frac{2}{5}=-\frac{4\sqrt{21}}{25}$
