Question

If the expression $\left(\right.1+tan x + \left(tan\right)^{2} ⁡ x \left.\right) \left(\right. 1 - cot ⁡ x + \left(cot\right)^{2} ⁡ x \left.\right)$ is positive, then the complete set of values of $x$ is

• A. $\left[0 , \frac{\pi }{2}\right]$
• B. $\left[0 , \pi \right]$
• C. $R-\left\{ \, x = \frac{n \pi }{2} , \, n \in I\right\}$
• D. $\left[0 , \, \in fty\right]$

Answer 1

Answer: (C)

$\frac{\left(1+\tan x+\tan ^{2} x\right)\left(1+\tan ^{2} x-\tan x\right)}{\tan ^{2} x}>0$
$\Rightarrow \frac{\left(1+\tan ^{2} x\right)^{2}-\tan ^{2} x}{\tan ^{2} x}>0$
Since, $1+tan^{2} x > tan^{2} ⁡ x , \, \forall x$ $\in R-\left\{x = \frac{n \pi }{2} , \, n \in I\right\}$
Hence, given expression is positive for all values of $x$ $\in R-\left\{x = \frac{n \pi }{2} , \, n \in I\right\}$