Answer :

We need to find the value of 2 tan-1 |cosec(tan-1 x) – tan(cot-1 x)|.

So, take


2 tan-1 |cosec(tan-1 x) – tan(cot-1 x)|


Using property of inverse trigonometry,





Now, let y = tan-1 x


So, tan y = x


Substituting the value of tan-1 x and x in the equation,



Put,






Since, we know the trigonometric identity,


1 – cos 2y = 2 sin2 y



Also, sin 2y = 2 sin y cos y



We get,




Since,



Then,




2 tan-1 |cosec(tan-1 x) – tan(cot-1 x)| = y


Put y = tan-1 x as let above.


2 tan-1 |cosec(tan-1 x) – tan(cot-1 x)| = tan-1 x

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