Choose the correct answer

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 sin^{2} 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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