Question

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

Answer 1

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.