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Prove that tan(cot–1x) = cot(tan–1x). State with reason whether the equality is valid for all values of x. - Mathematics

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Question

Prove that tan(cot–1x) = cot(tan–1x). State with reason whether the equality is valid for all values of x.

Sum
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Solution

Let cot–1x = θ.

Then cot θ = x

or

`tan(pi/2 - theta)` = x

⇒ `tan^-1x = pi/2 - theta`

So tan(cot–1x) = tan θ

= `cot(pi/2 - theta)`

= `cot(pi/2 - cot^-1 x)`

= cot(tan–1x)

The equality is valid for all values of x since tan–1x and cot–1x are true for x ∈ R.

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Chapter 2: Inverse Trigonometric Functions - Solved Examples [Page 22]

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NCERT Exemplar Mathematics [English] Class 12
Chapter 2 Inverse Trigonometric Functions
Solved Examples | Q 8 | Page 22

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