Solution:

Consider

$$\begin{aligned} 2 x=& x+y+x-y \ \tan 2 x &=\tan (x+y+x-y) \ &=\frac{\tan (x+y)+\tan (x-y)}{1-\tan (x+y) \tan (x-y)} \ &=\frac{\frac{3}{4}+\frac{8}{15}}{1-\frac{3}{4} \frac{8}{15}} \ &=\frac{77}{36} \ \therefore \tan 2 x &=77 / 36 \end{aligned}$$