Prove identity sec^-1(x) = cos^-1(1/x)

  • #1

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

1/cos(cos^-1(1/x)) = 1/1/x = x

cos^-1(1/x) = sec^-1(x).

Is this correct?
 

Answers and Replies

  • #2
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7,140

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
How do you justify the previous step?
1/cos(cos^-1(1/x)) = 1/1/x = x

cos^-1(1/x) = sec^-1(x).

Is this correct?
 
  • #3
HallsofIvy
Science Advisor
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The fundamental problem with try to prove "[itex]sec^{-1}(x)= cos^{-1}(1/x)[/itex]" is that it is NOT true! This is NOT an identity. For example, if [itex]x= \pi/4[/itex] [itex]sec^{-1}(x)= \sqrt{2}/2[/itex] while [itex]1/x= 4/\pi[/itex], cos(1/x)= 3.41, approximately.
 
  • #4
SammyS
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The fundamental problem with try to prove "[itex]sec^{-1}(x)= cos^{-1}(1/x)[/itex]" is that it is NOT true! This is NOT an identity. For example, if [itex]x= \pi/4[/itex] [itex]sec^{-1}(x)= \sqrt{2}/2[/itex] while [itex]1/x= 4/\pi[/itex], cos(1/x)= 3.41, approximately.
[itex]\displaystyle \sec(\pi/4)=\sqrt{2}\ [/itex]

[itex]\displaystyle \sec^{-1}(\pi/4)\ [/itex] is undefined, since [itex]\displaystyle\ \ \pi/4<1\ .[/itex]
 
  • #5
SammyS
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I would be inclined to write sec-1(x) as [itex]\displaystyle \ \sec^{-1}\left(\frac{1}{1/x}\right)\,,\ [/itex] then use the identity, cos(cos-1(u) = u , for the denominator.
 

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