Prove that:   1+e^z ---------=e^z 1+e^-z^ denotes power

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll have to prove the identity (1+e^z)/(1+e^-z) = e^z

We'll manage the left side and we'll get:

(1+e^z)/(1+e^-z) = (1+e^z)/(1+1/e^z)

We notice that we've used the negative power property to write the term e^-z = 1/e^z

We'll multiply 1 by e^z, within the brackets from denominator:

(1+e^z)/[(e^z + 1)/e^z]

The fraction from denominator [(e^z + 1)/e^z] will be reversed, such as:

(1+e^z)*e^z/(e^z + 1)

We'll simplify by (e^z + 1) and we'll get:

(1+e^z)*e^z/(e^z + 1) = e^z

We notice that managing the left side, we'll get the expression form the right side, therefore, the identity (1+e^z)/(1+e^-z) = e^z is verified.

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