tan –1 (tan θ ) = 
, sin –1 (sin θ ) = 
,
cos –1 (cos θ ) = 
Based on the above results, prove each of the following :
Based on the above results, prove each of the following :
(i) cos –1 x = sin –1 
if 0 < x < 1
(ii) sin –1 x = cos –1 
if 0 < x < 1
(iii) cos –1 x = π + tan –1 
if –1 < x < 0
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
(i) Let cos –1 x = θ , then x = cos θ and 0 ≤ θ ≤ π
∴ sin –1 
= sin –1 (sin θ ) = 
= 
∴ cos –1 x = sin –1 
if 0 < x < 1 is true.
(ii) Let sin –1 x = θ , then x = sin θ and – 
≤ θ ≤ 
∴ cos –1 
= cos –1 (cos θ ) = 
= 
sin –1 x = cos –1 
if 0 < x < 1 is true
(iii) Let cos –1 x = θ , then x = cos θ and 0 ≤ θ ≤ π
∴ tan –1 
= tan –1 (tan θ ) = 
= 
i.e cos –1 x = π + tan –1 
, –1 < x < 0 is correct.
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