Question

Differentiate the function given below w.r.t. $x:$

$(\sec x-1)(\sec x+1)$

Answer 1

Let $y=(\sec x-1)(\sec x+1)$

$y={\sec }^{2}x-1$ $[\because (a+b\left)\right(a-b)=a^2-b^2]$

$y={\tan }^{2}x$

Differentiate w.r.t. $x$

$\Rightarrow \frac{dy}{dx}=2\tan x\frac{d\left(\tan x\right)}{dx}$ $(\text{Chain rule)}$

$\Rightarrow \frac{dy}{dx}=2\tan x\left({\sec }^{2}x\right)$

$\Rightarrow \frac{dy}{dx}=2{\sec }^{2}x\tan x$