Question

If $f ( x )=\tan x +2 \tan ^2 x +3 \tan ^3 x +4 \tan ^4 x +\ldots \ldots \infty$ where $x \in\left(\frac{-\pi}{4}, \frac{\pi}{4}\right) \cup\left(\frac{3 \pi}{4}, \frac{5 \pi}{4}\right)$, then number of solution(s) of the equation $f(x)=2$, is(are)

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (B)

$ f ( x )=\frac{\tan x }{(1-\tan x )^2}=2$
$\Rightarrow 2\tan ^2 x -5 \tan x +2=0$
$\Rightarrow \tan x =\frac{1}{2}, 2 \text { (rejected) }$
$\therefore $ Number of solutions are 2 .