Answer: 150

Small Hint:

Express the sum of the cotangents using $\tan x$ and $\tan y$

Big Hint:

Use the tangent addition formula after finding $\tan x\tan y$

Solution:

Since $$\cot x+\cot y =\frac{\tan x+\tan y}{\tan x\tan y},$$ the given equations yield $\tan x\tan y=\frac{25}{30}=\frac{5}{6}.$ Therefore $$\begin{aligned} \tan(x+y) &=\frac{\tan x+\tan y} {1-\tan x\tan y}\\ &=\frac{25}{1-\frac56}\\ &=150. \end{aligned}$$