Question

The solution set of the equation $\tan (\pi \tan x)=\cot (\pi \cot x)$ is

• A. $\varphi$
• B. {0}
• C. $\left\{\frac{\pi }{4}\right\}$
• D. none of these.

Answer 1

Answer: (A)

$\tan (\pi \tan x)=\cot (\pi \cot x)=\tan \left(\frac{\pi }{2}=\pi \cot x\right)$
$\Rightarrow$ $\pi \tan x=\frac{\pi }{2}-\pi \cot x$
$\Rightarrow$ $\tan x+\cot x=\frac{1}{2}$
$\Rightarrow$ $\tan x+\cot x=\frac{1}{2}\tan x+\frac{1}{\tan x}=\frac{1}{2}\Rightarrow 2{\tan }^{2}x+2=\tan x$
$\Rightarrow$ $2{\tan }^{2}x-\tan x+2=0$
This is quadratic in $\tan x$
$\Rightarrow$ $\tan x=\frac{1\pm \sqrt{1-4.4}}{4}=\frac{1\pm \sqrt{-15}}{2}$
which are not real solution set = $\varphi$