Question

If $x$ and $y$ are real numbers such that $\tan x+\tan y=42$ and $\cot x+\cot y=49$ for all the permissible value of $x,y$, then the least prime number by which the value of $\tan (x+y)$ is not divisible is

Answer 1

$\tan x+\tan y=42\cot x+\cot y=49=\frac{1}{\tan x}+\frac{1}{\tan y}\Rightarrow \frac{\tan x+\tan y}{\tan x\tan y}=49\Rightarrow \frac{42}{\tan x\tan y}=49\Rightarrow \tan x\tan y=\frac{42}{49}=\frac{6}{7}\Rightarrow \tan (x+y)=\frac{\tan x+\tan y}{1-\tan x\tan y}\Rightarrow \tan (x+y)=\frac{42}{1-\frac{6}{7}}\Rightarrow \tan (x+y)=294=2\times 3\times 7^2$

Hence, the smallest prime number by which $\tan (x+y)$ is not divisible is $5$.