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Prove that:

cot -1 (√(1 + x2) – x) = π/2 – ½ cot -1 x

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Best answer

cot -1 (√(1 + x2) – x) = π/2 – ½ cot -1 x

Take x = cot θ

Where θ = cot -1 x

Here LHS = cot -1 (√(1 + x2) – x)

By substituting the value of x

= cot -1 (√(1 + cot2 θ) – cot θ)

It can be written as

= cot -1 (cosec θ – cot θ)

So we get

= cot -1 (1/sin θ – cos θ/sin θ)

On further calculation

= cot -1 ((1 – cos θ)/ sin θ)

We get

= cot -1 (2sin2 θ/2 / 2sin θ/2.cos θ/2)

By simplification

= cot -1 (tan θ/2)

Here,

= cot -1 (cot [π/2 – 1/2 θ])

So

= π/2 – 1/2 θ

By substituting the value of θ

= π/2 – 1/2 cot -1 x

= RHS

Hence, it is proved.

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