Q. The expression $(1 + \tan{x + \tan^{2}{x)(1 - \cot{x + \cot^{2}{x)}}}}$ has the positive value for $x$ given by
A) $0 \leq x \leq \frac{\pi}{2}$
B) $0 \leq x \leq \pi$
C) For all $x \in R$
D) $x \geq 0$
Explanation
(c)
$\frac{1 + \tan{x + \tan^{2}{x)(1 + tan^{2}x - \tan{x)}}}}{\tan^{2}x}$
$= \frac{(1 + {\tan^{2}{x)}}^{2} - \tan^{2}x}{\tan^{2}x}$
Since, $1 + \tan^{2}{x \geq \tan^{2}{x,\ \forall x}}$. Hence, it is positive for all values of $x$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.