Prove each of the following relations :
(i)tan–1 x = – π + cot–1 
= sin–1 
= – cos–1 
when x < 0.
(ii)cos–1x = sec–1 
= π – sin–1 
= π + tan–1 
= cot –1 
when tc – 1<x< 0
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
(i) Let tan–1 x = θ⇒ tan θ = x cot θ = 
x > 0
θ = – π + cot–1 
x < 0
sin θ = 
⇒θ = sin–1 
cos θ = 
x > 0
θ = cos–1 
= tan–1 x x > 0
and for x < 0 ⇒ cos–1 cos θ = cos–1 
⇒ – θ = cos–1 
⇒ tan–1x = –cos–1 
⇒ tan–1 x = – π + cot–1 
= sin–1 
= – cos–1 
where x < 0
(ii) Let
θ = cos–1 x given–1 < x < 0 ⇒ cos θ = x θ ∈ ( 
, π )
sec θ = 
θ = sec–1 
sin θ = 
⇒ θ = π – sin–1 
tan θ = 
⇒θ = π + tan–1 
cot θ = 
⇒ θ = cot–1 
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