Here in this post, learn how to draw the graph of $csc^(-1)(csc(x))$ step-by-step with clear instructions and examples in this comprehensive guide.

Here, we will understand how to draw the graph of cosec inverse cosec x i.e

$f(x) = cosec^{-1} cosec (x)$

As $cosec (\pi +x ) = cosec x$

This is a periodic function with period $2\pi$

Also, x cannot take values like $n\pi$ as undefined

We know by definition

$f(x) = cosec^{-1} cosec (x) = x $ if $x \in (-\pi/2, \pi/2) – {0}$

Let’s check out the other values

if $x \in (\pi/2, 3\pi/2] – {\pi}$

$\pi/2 < x \leq 3\pi/2 $

$ – 3\pi/2 \leq -x < -\pi/2$

Adding $\pi$

$ -\pi/2 \leq \pi – x < \pi/2 $

Also $cosec (\pi -x) = cosec x$

So $f(x) = cosec^{-1} cosec (x) = cosec^{-1} cosec ( \pi – x)= \pi -x$

here $ x \ne \pi$

if $x \in (3\pi/2, 2\pi)$

$3\pi/2 < x < 2\pi $

subtracting $2\pi$

$ -\pi/2 < x -2 |pi < 0 $

Also $cosec (2\pi -x) = -cosec x$

or $ -cosec (x -2 \pi) = -cosec x$

or $cosec (x-2 \pi)= cosec x$

So $f(x) = cosec^{-1} cosec (x) = cosec^{-1} cosec ( x -2 \pi)= x – \2pi$

here $ x \ne 2\pi$

Therefore the graph will be