Question

Suppose $x$ and $y$ are real numbers such that $\tan x+\tan y=42$ and $\cot x+\cot y=49$, then the single digit prime number by which the value of $\tan (x+y)$ is not divisible is

Answer 1

Given, $\tan x+\tan y=42$ and $\cot x+\cot y=49$

Let $\tan x=a$ and $\tan y=b$

So $a+b=42$ and $\frac{a+b}{ab}=49$

$ab=\frac{42}{49}$

$\tan (x+y)=\frac{\tan x+\tan y}{1-\tan x\tan y}=\frac{a+b}{1-ab}=\frac{42}{1-\frac{42}{49}}=42\times 7$

$\tan (x+y)=2\times 3\times 7\times 7$

So it is not divisible By $5$