If x ∈ ( 2 3 π , 2 π ) , then the value of tan − 1 ( tan x ) + cot − 1 ( cot x ) + sec − 1 ( sec x ) is:

Question

If x ∈ ( 2 3 π ​ , 2 π ) , then the value of tan − 1 ( tan x ) + cot − 1 ( cot x ) + sec − 1 ( sec x ) is:

TestHub · Accepted Answer

Given x ∈ ( 2 3 π ​ , 2 π ) . 1. For tan − 1 ( tan x ) : The principal value branch of tan − 1 is ( − π /2 , π /2 ) . Since x ∈ ( 2 3 π ​ , 2 π ) , we have x − 2 π ∈ ( − 2 π ​ , 0 ) . Also, tan ( x − 2 π ) = tan x . Thus, tan − 1 ( tan x ) = x − 2 π . 2. For cot − 1 ( cot x ) : The principal value branch of cot − 1 is ( 0 , π ) . Since x ∈ ( 2 3 π ​ , 2 π ) , we have x − π ∈ ( 2 π ​ , π ) . Also, cot ( x − π ) = cot x . Thus, cot − 1 ( cot x ) = x − π . 3. For sec − 1 ( sec x ) : The principal value branch of sec − 1 is [ 0 , π ] − { π /2 } . Since x ∈ ( 2 3 π ​ , 2 π ) , sec x > 0 . We have 2 π − x ∈ ( 0 , 2 π ​ ) . Also, sec ( 2 π − x ) = sec x . Thus, sec − 1 ( sec x ) = 2 π − x . Adding the three terms: ( x − 2 π ) + ( x − π ) + ( 2 π − x ) = x − π .

Mathematics - Inverse Trigonometric Functions Question with Solution | TestHub

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