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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Prove the identity.

13. coth2x – 1 = csch2x

To determine

To prove: The identity coth2x1=csch2x.

Explanation

Formula used:

The definition of the hyperbolic function are

(1) cschx=1sinhx.

(2) sinhx=exex2

(3) cothx=1tanhx

(4) tanhx=exexex+ex

Proof:

The left hand side of the identity coth2x1 can be expressed as follows,

coth2x1=1tanh2x1=1(exexex+ex)21=(ex+ex)2(exex)21=(ex+ex)2(exex)2(exex)2

Simplify further,

coth2x1=(e2x+e2x+2exex)(e</

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