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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
