Prove the following equation by using the method of this example.
[tex]\tanh ^{-1}(x)=\frac{1}{2} \ln \left(\frac{1+x}{1-x}\right) \quad-1
Answer (2)
Let y = tanh − 1 ( x ) , then x = tanh ( y ) .
Express tanh ( y ) in terms of exponentials: tanh ( y ) = e 2 y + 1 e 2 y − 1 .
Solve for e 2 y : e 2 y = 1 − x 1 + x .
Solve for y : y = 2 1 ln ( 1 − x 1 + x ) . Thus, tanh − 1 ( x ) = 2 1 ln ( 1 − x 1 + x ) .
Explanation
Problem Setup We are asked to prove the identity tanh − 1 ( x ) = 2 1 ln ( 1 − x 1 + x ) for − 1 < x < 1 . We will follow the method outlined in the provided example.
Substitution Let y = tanh − 1 ( x ) . Then, by definition, x = tanh ( y ) .
Expressing tanh(y) in terms of exponentials We express tanh ( y ) in terms of exponential functions. Recall that tanh ( y ) = c o s h ( y ) s i n h ( y ) , where sinh ( y ) = 2 e y − e − y and cosh ( y ) = 2 e y + e − y . Therefore, we have:
tanh ( y ) = 2 e y + e − y 2 e y − e − y = e y + e − y e y − e − y
To simplify this expression, we multiply the numerator and denominator by e y :
tanh ( y ) = e y ( e y + e − y ) e y ( e y − e − y ) = e 2 y + 1 e 2 y − 1
Solving for e^(2y) Now we set x = tanh ( y ) = e 2 y + 1 e 2 y − 1 and solve for y in terms of x .
x = e 2 y + 1 e 2 y − 1
Multiply both sides by ( e 2 y + 1 ) :
x ( e 2 y + 1 ) = e 2 y − 1
Expand the left side:
x e 2 y + x = e 2 y − 1
Rearrange the equation to isolate terms with e 2 y :
e 2 y − x e 2 y = x + 1
Factor out e 2 y :
e 2 y ( 1 − x ) = 1 + x
Divide by ( 1 − x ) to solve for e 2 y :
e 2 y = 1 − x 1 + x
Solving for y Take the natural logarithm of both sides:
2 y = ln ( 1 − x 1 + x )
Divide by 2 to solve for y :
y = 2 1 ln ( 1 − x 1 + x )
Final Result Since we initially set y = tanh − 1 ( x ) , we can substitute this back into the equation:
tanh − 1 ( x ) = 2 1 ln ( 1 − x 1 + x )
This completes the proof for − 1 < x < 1 .
Examples
The hyperbolic tangent inverse function is useful in various fields such as physics and engineering. For example, it appears in the study of rapid single-flux quantum (RSFQ) logic, which is a superconducting digital electronics technology. Also, it can be used to calculate the rapidity of a particle in special relativity, given its velocity. Understanding and manipulating such functions allows engineers and physicists to model and analyze complex systems more effectively.
We prove the equation tanh − 1 ( x ) = 2 1 ln ( 1 − x 1 + x ) by expressing x as tanh ( y ) and manipulating the equation to solve for y . After applying logarithms and algebraic rearrangements, we establish the identity. This proof highlights the relationships between hyperbolic functions and logarithms, essential in mathematics.
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