Trigonometry#0

Since $\tan x + \cot y = a$ , then multiplying through by $\tan y$ gives $\tan x \tan y = a \tan y - 1$ , and similarly for $\cot x + \tan y = b$ , multiplying through by $\tan x$ gives, $\tan x \tan y = b \tan x - 1$ . Therefore, $a \tan y = b \tan x$ , from which

$\dfrac{\tan y}{ \tan x} = \dfrac{b }{ a}$ .