Question

If a is any real number, the number of roots of $\cot x-\tan x=a$ in the first quardrant is

• A. $2$
• B. $0$
• C. $1$
• D. $4$

Answer 1

Answer: (C)

$1$

let $f\left(x\right)=\cot x-\tan x$ and $g\left(x\right)=a$
$f^{\prime }\left(x\right)=-({cosec}^{2}x+{\sec }^{2}x)<0$ in first quadrant
so $f\left(x\right)$ is a one one function
so $f\left(x\right)$ and $g\left(x\right)$ will intersect at only one point $(x,a)$
so $f\left(x\right)=a$ has only $1$ solution