Question

If a is any real number, the number of roots of $\cot x-\tan x=a$ in the first quadrant is (are).
(a) 2
(b) 0
(c) 1
(d) none of these

Answer 1

(c) 1

Given:

$$\begin{gathered} cotx-\tan x=a \\ \Rightarrow \frac{1}{\tan x}-\tan x=a \\ \Rightarrow 1-{\tan }^{2}x=a\tan x \\ \Rightarrow {\tan }^{2}x+a\tan x-1=0 \end{gathered}$$

It is a quadratic equation.
If $\tan x=z$, then the equation becomes
$z^2+az-1=0$
$$\begin{gathered} \Rightarrow z=\frac{-a\pm \sqrt{a^2+4}}{2} \\ \Rightarrow \tan x=\frac{-a\pm \sqrt{a^2+4}}{2} \\ \Rightarrow x={\tan }^{-1}\left(\frac{-a\pm \sqrt{a^2+4}}{2}\right) \end{gathered}$$
There are two roots of the given equation, but we need to find the number of roots in the first quadrant.
There is exactly one root of the equation, that is, $x={\tan }^{-1}\left(\frac{-a+\sqrt{a^2+4}}{2}\right)$.