Price the hyperbolic function 1-tanh^2x=sech^2x
Prove not price.
Prove the hyperbolic function 1-tanh^2x=sech^2x
cosh^2 - sinh^2 = 1
divide by cosh^2 to get
1 - tanh^2 = sech^2
tanh = sinh/cosh
sech = 1/cosh
1 - tanh^2 = 1 - sinh^2/cosh^2
sech^2 = 1/cosh^2
so
does
cosh^2 - sinh^2 = 1 ? Yes
in case you don't believe the basic hyperbolic trig identity
cosh^2 x - sinh^2 x = 1, go back to the definition.
((e^x + e^-x)/2)^2 - ((e^x - e^-x)/2)^2
= (e^2x + 2 + e^-2x)/4 - (e^2x - 2 + e^-2x)/4
= 4/4
= 1
To find the relationship between the hyperbolic functions in the equation 1 - tanh^2x = sech^2x and determine the value of x for which the equation holds true, follow these steps:
Step 1: Rewrite the equation using the definitions of the hyperbolic functions.
Recall that tanh(x) can be expressed as sinh(x)/cosh(x) and sech(x) as 1/cosh(x). Therefore, the equation can be rewritten as: 1 - sinh^2(x)/cosh^2(x) = 1/cosh^2(x).
Step 2: Find a common denominator.
Multiply both sides of the equation by cosh^2(x) to eliminate the denominators: cosh^2(x) - sinh^2(x) = 1.
Step 3: Use the hyperbolic trigonometric identities.
Apply the identity cosh^2(x) - sinh^2(x) = 1, which holds true for any real value of x. Therefore, the equation simplifies to 1 = 1.
Step 4: Analyze the result and interpret the solution.
Since the equation 1 = 1 is true for all values of x, you can conclude that the given equation 1 - tanh^2x = sech^2x is an identity. It holds true for all real values of x.
Hence, the price of the hyperbolic function 1 - tanh^2x = sech^2x is that it is an identity and holds true for all real values of x.