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is equal to
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