First of all, we have
[tex]\sin^2(u)+\cos^2(u)=1 \implies \cos(u)=\pm\sqrt{1-\sin^2(u)}[/tex]
Since u lies in the 4th quadrant, the cosine is positive, so we have
[tex]\cos(u)=\sqrt{1-\dfrac{9}{25}}=\dfrac{4}{5}[/tex]
Now, the double angles formula: we have
[tex]\sin(2u)=2\sin(u)\cos(u)=2\cdot\left(-\dfrac{3}{5}\right)\cdot \dfrac{4}{5}=-\dfrac{24}{25}[/tex]
[tex]\cos(2u)=cos^2(u)-\sin^2(u)=\dfrac{16}{25}-\dfrac{9}{25}=\dfrac{7}{25}[/tex]
For the tangent, we can simply use the definition:
[tex]tan(2u)=\dfrac{\sin(2u)}{\cos(2u)}=\dfrac{-\frac{24}{25}}{\frac{7}{25}}=-\dfrac{24}{7}[/tex]
