Question

Prove the hyperbolic function 1−tanh^2 x=sech^2 x.

Answer 1

To prove 1−tanh^2 x = sech^2 x, we start with the definitions of the hyperbolic functions and manipulate them using identities.

Step-by-step explanation:

To prove the hyperbolic function 1−tanh^2 x=sech^2 x, we start with the definition of the hyperbolic functions:

tanh x = sinh x / cosh x

sech x = 1 / cosh x

We square both sides of the second equation to get:

sech^2 x = (1 / cosh x)^2 = 1 / cosh^2 x

Next, we use the identity cosh^2 x = 1 + sinh^2 x, which can be derived from the Pythagorean identity for the cosine and sine functions:

cosh^2 x - sinh^2 x = 1

Simplifying, we find 1 - tanh^2 x = sech^2 x, which completes the proof.