The discussion revolves around proving the identity sec-1(x) = cos-1(1/x). Participants are exploring the relationship between the secant and cosine inverse functions, questioning the validity of the proposed identity.
The discussion includes attempts to prove the identity, with some participants expressing skepticism about its validity. There is a recognition of potential errors in reasoning, and participants are exploring different interpretations of the functions involved.
Participants note that the identity may not be valid for all values of x, particularly highlighting cases where x is less than 1, which raises questions about the definitions and domains of the functions involved.
How do you justify the previous step?physicsernaw said:Homework Statement
Find and prove the identity sec^-1(x) in terms of cos^-1(arg) (Note that 1/cos^-1(x) is not equal to sec^-1(x).
Homework Equations
None.
The Attempt at a Solution
sec(sec^-1(x)) = x
1/cos(sec^-1(x)) = x
1/cos(cos^-1(x)) = 1/x
physicsernaw said:1/cos(cos^-1(1/x)) = 1/1/x = x
cos^-1(1/x) = sec^-1(x).
Is this correct?
\displaystyle \sec(\pi/4)=\sqrt{2}\HallsofIvy said:The fundamental problem with try to prove "sec^{-1}(x)= cos^{-1}(1/x)" is that it is NOT true! This is NOT an identity. For example, if x= \pi/4 sec^{-1}(x)= \sqrt{2}/2 while 1/x= 4/\pi, cos(1/x)= 3.41, approximately.