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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.
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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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