Question

The equation of normal to tha curve $y=\tan x$ at $(0,0)$ is .

Answer 1

Given: $y=\tan x$
$\Rightarrow \frac{dy}{dx}={\sec }^{2}x$
$\Rightarrow {\frac{dy}{dx}|}_{(0,0)}={\sec }^{2}0=1$
Slope of normal at $(0,0)$
$=\frac{-1}{\text{slope of tangent at (0,0)}}=-1$
Line passing through $(0,0)$ with slope -1,
$\Rightarrow y-0=-1(x-0)$
$[\therefore y-y_1=m(x-x_1]$
$\Rightarrow x+y=0$
Hence, the equation of normal to curve
$y=\tan x\text{at}(0,0)\text{is}x+y=0$