Question

The expression $($1 $+$ tan x $+{\tan }^{2}x\left)\right(1-cotx+{\cot }^{2}x)$ has a value $\ge 3$

• A. $0\le x\le \frac{\pi }{2}$
• B. $0\le x\le \pi$
• C. $x\in R,\forall x$
• D. $\forall x\in R$ excepting $x=\frac{n\pi }{2},n\in Z$

Answer 1

The correct option is D $\forall x\in R$ excepting $x=\frac{n\pi }{2},n\in Z$

$f\left(x\right)=(1+tanx+ta{n}^{2}x)(1-cotx+co{t}^{2}x)$
$=1-cotx+co{t}^{2}x+tanx-1+cotx+ta{n}^{2}x-tanx+1$
$f\left(x\right)=1+co{t}^{2}x+ta{n}^{2}x$
take two +ve numbers $co{t}^{2}x,ta{n}^{2}x$
$\therefore$ By A.M - G.M. inequality
$\frac{co{t}^{2}x+ta{n}^{2}x}{2}\ge (ta{n}^{2}xco{t}^{2}x{)}^{\frac{1}{2}}$
$co{t}^{2}x+ta{n}^{2}x\ge 2$
$f\left(x\right)=1+co{t}^{2}x+ta{n}^{2}x\ge 3$
This is True for $xϵR$ except $x=\frac{4\pi }{2},xϵz$ because at $x=\frac{4\pi }{2},tanx$ is not define.