then both sides ls & rs expand to:
Hello!
I was asked to show that the following identity holds:
|cosh(z)|^{2}= sinh^{2}(x) + cos^{2}(y)
This is what I've got,
LS= |cosh(z)|^{2 } = |1/2(e^{z}+e^{-}^{z})|^{2}
=1/4(e^{z}+e^{-}^{z})(e^{z}+e^{-}^{z})
=1/4(e^{2}^{z}+2+e^{-2}^{z})RS= sinh^{2}(x) + cos^{2}(y)
= (1/2)(e^{x}-e^{-}^{x})(1/2)(e^{x}-e^{-}^{x})
+ (1/2)(e^{iy}+e^{-}^{iy})(1/2)(e^{iy}+e^{-}^{iy})
= 1/4(e^{2x}-2+e^{-2x}+e^{2iy}+2+e^{-2iy})
= 1/4(e^{2}^{z}+e^{-2}^{z})
So LS does not equal RS....
Where is my mistake? Is it in the Absolute value set on the LS? that's the only part that is not intuitive for me...
Any hints would be killler!!!
Thanks MaxJasper,
I've never seen this expansion of |cosh(z)| before... and not quite sure if I understand it.
Could you maybe explain the step from left side to right?
I've only seen |cosh(z)|= (cosh(z)cosh(z))^{1/2}
I would really appreciate that!
Edit:
Are you using trig identities?
Such as
cosh(z)=cos(iz)
cos(iz)= cos(ix)cosh(iy)- (i)sin(ix)sinh(iy)
Isn't |cosh(z)|= (cosh(z)cosh(z))^{1/2 }a legitimate expansion of |cosh(z)|. So shouldn't it work?