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Chapter 2: Q. 98 (page 236)

Prove that the inverse hyperbolic functions can be written in terms of logarithms as shown in Exercises 97鈥�99.

cosh-1x=lnx+x2-1forx鈮�1.

Short Answer

Expert verified

We proved cosh-1x=lnx+x2-1for x鈮�1.

Step by step solution

01

Step 1. Given Information

We have cosh=xand we need to prove cosh-1x=lnx+x2-1.

02

Step 2. Proving the statement 

coshy=x鈬�ey+e-y2=x鈬�ey+e-y=2x鈬�ey+1ey=2x鈬�e2y+1ey=2x鈬�e2y+1=2xey鈬�e2y-2xey+1=0

Solving the quadratic we get,

ey=2x卤4x2-42鈬�lney=ln2x卤4x2-42鈬�lney=ln2x卤x2-12鈬�y=lnx卤x2-1鈬�cosh-1x=lnx卤x2-1

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