Find principal value of tan1 (tan 2π÷3)
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The text version of the answer is 
Find the principal value of tan1(tan〖2π/3〗 )
Let y = tan1(tan〖2π/3〗 )
tan y =〖 tan〗〖2π/3〗
tan y = tan (120°)
We know that range of principal value of tan1 is
(− π/2 , π/2 ) i.e. (− 90° ,90°)
Hence y = 120° not possible
Now,
tan y = tan (120°)
tan y = tan (180° – 60°)
tan y = – tan (60°)
tan y = tan (–60°)
tan y = tan (–60 × π/180)
tan y = tan ((π)/3)
Hence, y = (π)/3
Which is in the range of principal value of tan1 i.e. ((π)/2, π/2)
Hence, tan1(tan〖2π/3〗 ) = y = (  π)/ 3
Notes 
(As tan (180 – θ ) = – tan θ )
(As tan (– θ ) = – tan θ )
Range 

sin 1 
[π/2, π/2] 
cos 1 
[0,π] 
tan 1 
(π/2, π/2) 