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Wardlaw
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Show that cosh(2z)=cosh^2(z)+sinh^2(z)
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Wardlaw said:Show that cosh(2z)=cosh^2(z)+sinh^2(z)
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sjb-2812 said:Do you know Osborn's rule? ( http://en.wikipedia.org/wiki/Osborn's_Rule#Similarities_to_circular_trigonometric_functions )
Wardlaw said:Show that cosh(2z)=cosh^2(z)+sinh^2(z)
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Wardlaw said:Yeah. I tried using the standard form for these expressions, when considering the RHS. I am then left with a quarter e^2z. Could you check this please?
tiny-tim said:Hi Wardlaw!
(try using the X^{2} tag just above the Reply box )
You should get some e^{-2z} also.
Show us what you got for the RHS.
Wardlaw said:How exactly do you go about solving thi problem?
tiny-tim said:I leave it to you.
The equation states that the hyperbolic cosine of twice the variable z is equal to the sum of the square of the hyperbolic cosine of z and the square of the hyperbolic sine of z.
This equation is a fundamental relationship in hyperbolic trigonometry and is often used in solving problems involving hyperbolic functions.
The equation can be derived using the double angle formula for the hyperbolic cosine and the fundamental identity for hyperbolic functions.
The equation has various applications in fields such as physics, engineering, and mathematics. It is used in solving problems related to hyperbolic functions, such as calculating the area under a hyperbolic curve or finding the solution to differential equations.
Yes, the equation can also be written as cosh(2z) = 1 + 2sinh^2(z) or cosh(z)^2 - sinh(z)^2 = 1. These are all equivalent forms of the same fundamental relationship between hyperbolic functions.