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Problem 1. (a) Show that that 1 – tanh² x = sech² x and use it to prove that sech(tanh x) = v1 – x².

Problem 1. (a) Show that that 1 – tanh² x = sech² x and use it to prove that sech(tanh x) = v1 – x².

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Problem 1. (a) Show that that 1 – tanh? x = sech x and use it to prove that sech(tanh x) = v1 – x². (b) Show that tanh(x) = ; In () 1 1+ x -1 1- x d (c) Justify that Method 1. Differentiate the identity tanh(tanh (x)) 1 in two different ways. - x2 tanh- (x) dx 1 = x. 1+x Method 2. Differentiate the identity tanh¯(x) = , In 1- x
Transcribed Image Text:Problem 1. (a) Show that that 1 – tanh? x = sech x and use it to prove that sech(tanh x) = v1 – x². (b) Show that tanh(x) = ; In () 1 1+ x -1 1- x d (c) Justify that Method 1. Differentiate the identity tanh(tanh (x)) 1 in two different ways. - x2 tanh- (x) dx 1 = x. 1+x Method 2. Differentiate the identity tanh¯(x) = , In 1- x
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Since you have asked multiple questions, we will solve the first question for you. If you want any specific question to be solved then please specify the question number or post only that question.

(a.) We have to show that  1-tanh2x=sech2x and then using this we have to prove  sechtanh-1x=1-x2.

Now we know that 

sinh x=ex-e-x2

cosh x=ex+e-x2

tanh x=sinh xcosh x=ex-e-xex+e-x

sech x=1cosh x=2ex+e-x

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