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Sech^2(x) = 1 - tanh^2(x) proof - need help!

  • Thread starter Tui
  • Start date
  • #1
Tui
15
0

Homework Statement



Prove Sech^2(x) = 1 - tanh^2(x)

Homework Equations



TanH(x) = (e^x - e^-x)/(e^x+e^-x)
CosH(x) = (e^x+e^-x)/2
SinH(x) = (e^x - e^-x)/2

The Attempt at a Solution



SecH^2(x) = 1/cosh^2(x)
=1 / (e^x - e^-x)^2 / 4
=4/(e^x - e^-x)^2

This is where I am stuck. Any help is greatly appreciated. Thank you !
 

Answers and Replies

  • #2
1,082
25
I'd recommend proving a simpler statement which directly leads to the statement you want to prove.

Try proving Cosh^2(x)-sinh^2(x)=1
 
  • #3
392
17

Homework Statement



Prove Sech^2(x) = 1 - tanh^2(x)

Homework Equations



TanH(x) = (e^x - e^-x)/(e^x+e^-x)
CosH(x) = (e^x+e^-x)/2
SinH(x) = (e^x - e^-x)/2

The Attempt at a Solution



SecH^2(x) = 1/cosh^2(x)
=1 / (e^x - e^-x)^2 / 4
=4/(e^x - e^-x)^2

This is where I am stuck. Any help is greatly appreciated. Thank you !
Try working from the more complicated side and work towards the simpler side. Often when you do this, terms cancel somewhere. If you start from the simpler side you usually need to creatively add 0 or multiply by 1, and this is often not that easy to see. Simplifying is often easier to see.
 
  • #4
SammyS
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
11,315
1,005

Homework Statement



Prove Sech^2(x) = 1 - tanh^2(x)

Homework Equations



TanH(x) = (e^x - e^-x)/(e^x+e^-x)
CosH(x) = (e^x+e^-x)/2
SinH(x) = (e^x - e^-x)/2

The Attempt at a Solution



SecH^2(x) = 1/cosh^2(x)
=1 / (e^x - e^-x)^2 / 4
=4/(e^x - e^-x)^2

This is where I am stuck. Any help is greatly appreciated. Thank you !
Use you expression for tanh(x). Square that, then use a common denominator to combine 1 - tanh2(x) into one fraction.
 

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