Solve for y where y = e^(x^2) / (tan^2(x) + 3) and y = (e^(mx) + e^(-mx)) / (e^(mx) - e^(-mx))
Understand the Problem
The question is presenting two expressions for 'y' and likely it needs help simplifying or solving it. Let's format the expressions within \LaTeX\ for clarity and attempt to simplify it.
Answer
$y = \frac{e^{x^2}}{\tan^2{x} + 3}$ $y = \coth(mx)$
Answer for screen readers
$y = \frac{e^{x^2}}{\tan^2{x} + 3}$ $y = \coth(mx)$
Steps to Solve
- Recognize the hyperbolic tangent function
The second equation for $y$ resembles the definition of the hyperbolic tangent function, often written as $\tanh(x)$. Recall that $\tanh(x)$ is defined as:
$$ \tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}} $$
- Manipulate the given expression
The given expression for the second equation is:
$$ y = \frac{e^{mx} + e^{-mx}}{e^{mx} - e^{-mx}} $$ We can rewrite this as: $$ y = \frac{e^{mx} + e^{-mx}}{e^{mx} - e^{-mx}} = \frac{e^{mx} + e^{-mx}}{-(e^{-mx} - e^{mx})} = - \frac{e^{mx} + e^{-mx}}{e^{-mx} - e^{mx}} $$
$$y = \frac{e^{mx} + e^{-mx}}{e^{mx} - e^{-mx}}$$ Dividing the numerator and the denominator by $e^{-mx}$:
$$y = \frac{e^{2mx} + 1}{e^{2mx} - 1}$$ Also, note that: $$ \coth(x) = \frac{1}{\tanh(x)} = \frac{\cosh(x)}{\sinh(x)} = \frac{e^x + e^{-x}}{e^x - e^{-x}} $$ Therefore, we can express the second $y$ as: $$ y = \coth(mx) $$
- Summarize the simplified forms
We have the two expressions for $y$: $$ y = \frac{e^{x^2}}{\tan^2(x) + 3} $$ and $$ y = \coth(mx) $$ or $$ y = \frac{e^{2mx} + 1}{e^{2mx} - 1} $$
$y = \frac{e^{x^2}}{\tan^2{x} + 3}$ $y = \coth(mx)$
More Information
The hyperbolic cotangent, denoted as $\coth(x)$ is the reciprocal of the hyperbolic tangent function $\tanh(x)$.
Tips
A common mistake is not recognizing the hyperbolic functions and attempting to simplify the exponential terms incorrectly. Another mistake is not being careful with algebraic manipulations. Avoid these mistakes by carefully reviewing definitions and properties of hyperbolic functions and double-checking each step of the algebraic simplification.
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