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Popular Trigonometry >

(1+tanh(x))/(1-tanh(x))=2

  • Pre Algebra
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Solution

1−tanh(x)1+tanh(x)​=2

Solution

x=21​ln(2)
+1
Degrees
x=19.85720…∘
Solution steps
1−tanh(x)1+tanh(x)​=2
Rewrite using trig identities
1−tanh(x)1+tanh(x)​=2
Use the Hyperbolic identity: tanh(x)=ex+e−xex−e−x​1−ex+e−xex−e−x​1+ex+e−xex−e−x​​=2
1−ex+e−xex−e−x​1+ex+e−xex−e−x​​=2
1−ex+e−xex−e−x​1+ex+e−xex−e−x​​=2:x=21​ln(2)
1−ex+e−xex−e−x​1+ex+e−xex−e−x​​=2
Multiply both sides by 1−ex+e−xex−e−x​1−ex+e−xex−e−x​1+ex+e−xex−e−x​​(1−ex+e−xex−e−x​)=2(1−ex+e−xex−e−x​)
Simplify1+ex+e−xex−e−x​=2(1−ex+e−xex−e−x​)
Apply exponent rules
1+ex+e−xex−e−x​=2(1−ex+e−xex−e−x​)
Apply exponent rule: abc=(ab)ce−x=(ex)−11+ex+(ex)−1ex−(ex)−1​=2(1−ex+(ex)−1ex−(ex)−1​)
1+ex+(ex)−1ex−(ex)−1​=2(1−ex+(ex)−1ex−(ex)−1​)
Rewrite the equation with ex=u1+u+(u)−1u−(u)−1​=2(1−u+(u)−1u−(u)−1​)
Solve 1+u+u−1u−u−1​=2(1−u+u−1u−u−1​):u=2​,u=−2​
1+u+u−1u−u−1​=2(1−u+u−1u−u−1​)
Refine1+u2+1u2−1​=2(1−u2+1u2−1​)
Multiply both sides by u2+1
1+u2+1u2−1​=2(1−u2+1u2−1​)
Multiply both sides by u2+11⋅(u2+1)+u2+1u2−1​(u2+1)=2(1−u2+1u2−1​)(u2+1)
Simplify
1⋅(u2+1)+u2+1u2−1​(u2+1)=2(1−u2+1u2−1​)(u2+1)
Simplify 1⋅(u2+1):u2+1
1⋅(u2+1)
Multiply: 1⋅(u2+1)=(u2+1)=(u2+1)
Remove parentheses: (a)=a=u2+1
Simplify u2+1u2−1​(u2+1):u2−1
u2+1u2−1​(u2+1)
Multiply fractions: a⋅cb​=ca⋅b​=u2+1(u2−1)(u2+1)​
Cancel the common factor: u2+1=u2−1
u2+1+u2−1=2(1−u2+1u2−1​)(u2+1)
Simplify u2+1+u2−1:2u2
u2+1+u2−1
Group like terms=u2+u2+1−1
Add similar elements: u2+u2=2u2=2u2+1−1
1−1=0=2u2
2u2=2(1−u2+1u2−1​)(u2+1)
2u2=2(1−u2+1u2−1​)(u2+1)
2u2=2(1−u2+1u2−1​)(u2+1)
Expand 2(1−u2+1u2−1​)(u2+1):4
2(1−u2+1u2−1​)(u2+1)
Expand (1−u2+1u2−1​)(u2+1):2
(1−u2+1u2−1​)(u2+1)
Apply FOIL method: (a+b)(c+d)=ac+ad+bc+bda=1,b=−u2+1u2−1​,c=u2,d=1=1⋅u2+1⋅1+(−u2+1u2−1​)u2+(−u2+1u2−1​)⋅1
Apply minus-plus rules+(−a)=−a=1⋅u2+1⋅1−u2+1u2−1​u2−1⋅u2+1u2−1​
Simplify 1⋅u2+1⋅1−u2+1u2−1​u2−1⋅u2+1u2−1​:2
1⋅u2+1⋅1−u2+1u2−1​u2−1⋅u2+1u2−1​
1⋅u2=u2
1⋅u2
Multiply: 1⋅u2=u2=u2
1⋅1=1
1⋅1
Multiply the numbers: 1⋅1=1=1
u2+1u2−1​u2=u2+1u4−u2​
u2+1u2−1​u2
Multiply fractions: a⋅cb​=ca⋅b​=u2+1(u2−1)u2​
Expand (u2−1)u2:u4−u2
(u2−1)u2
=u2(u2−1)
Apply the distributive law: a(b−c)=ab−aca=u2,b=u2,c=1=u2u2−u2⋅1
=u2u2−1⋅u2
Simplify u2u2−1⋅u2:u4−u2
u2u2−1⋅u2
u2u2=u4
u2u2
Apply exponent rule: ab⋅ac=ab+cu2u2=u2+2=u2+2
Add the numbers: 2+2=4=u4
1⋅u2=u2
1⋅u2
Multiply: 1⋅u2=u2=u2
=u4−u2
=u4−u2
=u2+1u4−u2​
1⋅u2+1u2−1​=u2+1u2−1​
1⋅u2+1u2−1​
Multiply: 1⋅u2+1u2−1​=u2+1u2−1​=u2+1u2−1​
=u2+1−u2+1u4−u2​−u2+1u2−1​
Combine the fractions −u2+1u4−u2​−u2+1u2−1​:−u2+1
Apply rule ca​±cb​=ca±b​=u2+1−(u4−u2)−(u2−1)​
Factor −(u2−1)−(u4−u2):−(u2−1)(u2+1)
−(u2−1)−(u4−u2)
Factor u4−u2:u2(u2−1)
u4−u2
Factor out common term u2:u2(u2−1)
u4−u2
Apply exponent rule: ab+c=abacu4=u2u2=u2u2−u2
Factor out common term u2=u2(u2−1)
=u2(u2−1)
=−u2(u2−1)−(u2−1)
Factor out common term (u2−1)=(u2−1)(−u2−1)
Factor −u2−1:−(u2+1)
−u2−1
Factor out common term −1=−(u2+1)
=−(u2−1)(u2+1)
=−u2+1(u2−1)(u2+1)​
Cancel the common factor: u2+1=−(u2−1)
Negate −(u2−1)=−u2+1=−u2+1
=u2+1−u2+1
Group like terms=u2−u2+1+1
Add similar elements: u2−u2=0=1+1
Add the numbers: 1+1=2=2
=2
=2⋅2
Expand 2⋅2:4
2⋅2
Distribute parentheses=2⋅2
Multiply the numbers: 2⋅2=4=4
=4
2u2=4
Solve 2u2=4:u=2​,u=−2​
2u2=4
Divide both sides by 2
2u2=4
Divide both sides by 222u2​=24​
Simplifyu2=2
u2=2
For x2=f(a) the solutions are x=f(a)​,−f(a)​
u=2​,u=−2​
u=2​,u=−2​
Verify Solutions
Find undefined (singularity) points:u=0
Take the denominator(s) of 1+u+u−1u−u−1​ and compare to zero
u=0
Take the denominator(s) of 2(1−u+u−1u−u−1​) and compare to zero
u=0
The following points are undefinedu=0
Combine undefined points with solutions:
u=2​,u=−2​
u=2​,u=−2​
Substitute back u=ex,solve for x
Solve ex=2​:x=21​ln(2)
ex=2​
Apply exponent rules
ex=2​
Apply exponent rule: a​=a21​2​=221​ex=221​
If f(x)=g(x), then ln(f(x))=ln(g(x))ln(ex)=ln(221​)
Apply log rule: ln(ea)=aln(ex)=xx=ln(221​)
Apply log rule: ln(xa)=a⋅ln(x)ln(221​)=21​ln(2)x=21​ln(2)
x=21​ln(2)
Solve ex=−2​:No Solution for x∈R
ex=−2​
af(x) cannot be zero or negative for x∈RNoSolutionforx∈R
x=21​ln(2)
Verify Solutions:x=21​ln(2)True
Check the solutions by plugging them into 1−ex+e−xex−e−x​1+ex+e−xex−e−x​​=2
Remove the ones that don't agree with the equation.
Plug in x=21​ln(2):True
1−e21​ln(2)+e−21​ln(2)e21​ln(2)−e−21​ln(2)​1+e21​ln(2)+e−21​ln(2)e21​ln(2)−e−21​ln(2)​​=2
1−e21​ln(2)+e−21​ln(2)e21​ln(2)−e−21​ln(2)​1+e21​ln(2)+e−21​ln(2)e21​ln(2)−e−21​ln(2)​​=2
1−e21​ln(2)+e−21​ln(2)e21​ln(2)−e−21​ln(2)​1+e21​ln(2)+e−21​ln(2)e21​ln(2)−e−21​ln(2)​​
e21​ln(2)+e−21​ln(2)e21​ln(2)−e−21​ln(2)​=31​
e21​ln(2)+e−21​ln(2)e21​ln(2)−e−21​ln(2)​
e21​ln(2)=2​
e21​ln(2)
Apply exponent rule: abc=(ab)c=eln(2)​
Apply log rule: aloga​(b)=beln(2)=2=2​
e−21​ln(2)=2​1​
e−21​ln(2)
Apply exponent rule: abc=(ab)c=(eln(2))−21​
Apply log rule: aloga​(b)=beln(2)=2=2−21​
Apply exponent rule: a−b=ab1​=2​1​
=2​+2​1​e21​ln(2)−e−21​ln(2)​
e21​ln(2)=2​
e21​ln(2)
Apply exponent rule: abc=(ab)c=eln(2)​
Apply log rule: aloga​(b)=beln(2)=2=2​
e−21​ln(2)=2​1​
e−21​ln(2)
Apply exponent rule: abc=(ab)c=(eln(2))−21​
Apply log rule: aloga​(b)=beln(2)=2=2−21​
Apply exponent rule: a−b=ab1​=2​1​
=2​+2​1​2​−2​1​​
Join 2​+2​1​:2​3​
2​+2​1​
Convert element to fraction: 2​=2​2​2​​=2​2​2​​+2​1​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=2​2​2​+1​
2​2​+1=3
2​2​+1
Apply radical rule: a​a​=a2​2​=2=2+1
Add the numbers: 2+1=3=3
=2​3​
=2​3​2​−2​1​​
Join 2​−2​1​:2​1​
2​−2​1​
Convert element to fraction: 2​=2​2​2​​=2​2​2​​−2​1​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=2​2​2​−1​
2​2​−1=1
2​2​−1
Apply radical rule: a​a​=a2​2​=2=2−1
Subtract the numbers: 2−1=1=1
=2​1​
=2​3​2​1​​
Divide fractions: dc​ba​​=b⋅ca⋅d​=2​⋅31⋅2​​
Refine=2​⋅32​​
Cancel the common factor: 2​=31​
=1−31​1+e21​ln(2)+e−21​ln(2)e21​ln(2)−e−21​ln(2)​​
e21​ln(2)+e−21​ln(2)e21​ln(2)−e−21​ln(2)​=31​
e21​ln(2)+e−21​ln(2)e21​ln(2)−e−21​ln(2)​
e21​ln(2)=2​
e21​ln(2)
Apply exponent rule: abc=(ab)c=eln(2)​
Apply log rule: aloga​(b)=beln(2)=2=2​
e−21​ln(2)=2​1​
e−21​ln(2)
Apply exponent rule: abc=(ab)c=(eln(2))−21​
Apply log rule: aloga​(b)=beln(2)=2=2−21​
Apply exponent rule: a−b=ab1​=2​1​
=2​+2​1​e21​ln(2)−e−21​ln(2)​
e21​ln(2)=2​
e21​ln(2)
Apply exponent rule: abc=(ab)c=eln(2)​
Apply log rule: aloga​(b)=beln(2)=2=2​
e−21​ln(2)=2​1​
e−21​ln(2)
Apply exponent rule: abc=(ab)c=(eln(2))−21​
Apply log rule: aloga​(b)=beln(2)=2=2−21​
Apply exponent rule: a−b=ab1​=2​1​
=2​+2​1​2​−2​1​​
Join 2​+2​1​:2​3​
2​+2​1​
Convert element to fraction: 2​=2​2​2​​=2​2​2​​+2​1​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=2​2​2​+1​
2​2​+1=3
2​2​+1
Apply radical rule: a​a​=a2​2​=2=2+1
Add the numbers: 2+1=3=3
=2​3​
=2​3​2​−2​1​​
Join 2​−2​1​:2​1​
2​−2​1​
Convert element to fraction: 2​=2​2​2​​=2​2​2​​−2​1​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=2​2​2​−1​
2​2​−1=1
2​2​−1
Apply radical rule: a​a​=a2​2​=2=2−1
Subtract the numbers: 2−1=1=1
=2​1​
=2​3​2​1​​
Divide fractions: dc​ba​​=b⋅ca⋅d​=2​⋅31⋅2​​
Refine=2​⋅32​​
Cancel the common factor: 2​=31​
=1−31​1+31​​
Simplify
1−31​1+31​​
Join 1−31​:32​
1−31​
Convert element to fraction: 1=31⋅3​=31⋅3​−31​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=31⋅3−1​
1⋅3−1=2
1⋅3−1
Multiply the numbers: 1⋅3=3=3−1
Subtract the numbers: 3−1=2=2
=32​
=32​1+31​​
Join 1+31​:34​
1+31​
Convert element to fraction: 1=31⋅3​=31⋅3​+31​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=31⋅3+1​
1⋅3+1=4
1⋅3+1
Multiply the numbers: 1⋅3=3=3+1
Add the numbers: 3+1=4=4
=34​
=32​34​​
Divide fractions: dc​ba​​=b⋅ca⋅d​=3⋅24⋅3​
Cancel the common factor: 3=24​
Divide the numbers: 24​=2=2
=2
2=2
True
The solution isx=21​ln(2)
x=21​ln(2)

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Frequently Asked Questions (FAQ)

  • What is the general solution for (1+tanh(x))/(1-tanh(x))=2 ?

    The general solution for (1+tanh(x))/(1-tanh(x))=2 is x= 1/2 ln(2)
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