Question

The value of cot^–1(–x) for all x ∈ R in terms of cot^–1x is ______.

Answer 1

The value of cot^–1(–x) for all x ∈ R in terms of cot^–1x is π – cot^–1x.

Explanation:

Clearly, –x ∈ R for all  x ∈ R

Let cot^–1(–x) = θ, θ ∈ (0, π)   ......(i)

⇒  –x = cot θ

⇒ x = – cot θ

⇒ x = cot (π – θ)

⇒ cot^–1x = π – θ  .......[∵ x ∈ R and π – θ ∈ (0, π) for all θ ∈ (0, π)]

⇒ θ = π – cot^–1x   .....(ii)

From (i) and (ii), we get

cot^–1(–x) = π – cot^–1x