Question

If the expression $(1+tanx+{\left(tan\right)}^{2}x)(1-cotx+{\left(cot\right)}^{2}x)$ 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, $1+ta{n}^{2}x>ta{n}^{2}x,\forall x$ $\in R-\left\{x=\frac{n\pi }{2},n\in I\right\}$
Hence, given expression is positive for all values of $x$ $\in R-\left\{x=\frac{n\pi }{2},n\in I\right\}$