From Tan to Cot and back to Tan

$\cot x + \cot y = 30$ $\implies \frac{1}{\tan x}+\frac{1}{\tan y}=30$ $\implies \frac{\tan x + \tan y}{\tan x \tan y}=30$

We were given that $\tan x + \tan y=25$

$\implies \frac{25}{30}=\tan x \tan y$

Subtract both sides from 1: $\implies 1 - \tan x \tan y = \frac{5}{30}$

Divide both side by $\tan x + \tan y=25$ $\implies \frac{\tan x + \tan y}{ 1 - \tan x \tan y} = 25 \times \frac{30}{5}$

But then the LHS is now $\tan (x+y)$ !

$\boxed{\tan(x+y)=150}$

By the way, to write $\tan x$ please use

\tan x

for better formatting