Sure, let's solve the expression step by step:
We want to simplify or evaluate the given expression [tex]\(\frac{\sin 2 x}{x + \cosh 2 x}\)[/tex].
1. Identify the components of the expression:
- Numerator: [tex]\(\sin(2x)\)[/tex]
- Denominator: [tex]\(x + \cosh(2x)\)[/tex]
2. Understand each part:
- [tex]\(\sin(2x)\)[/tex]: This is the sine function, which is a periodic trigonometric function that varies between -1 and 1. The argument here is [tex]\(2x\)[/tex] instead of [tex]\(x\)[/tex].
- [tex]\(\cosh(2x)\)[/tex]: This is the hyperbolic cosine function. [tex]\(\cosh(2x) = \frac{e^{2x} + e^{-2x}}{2}\)[/tex]. It always gives positive values and grows exponentially as [tex]\(x\)[/tex] increases.
3. Combine the components in the fraction:
Now we combine the two parts into our original fraction:
[tex]\[
\frac{\sin(2x)}{x + \cosh(2x)}
\][/tex]
This represents the entire expression given.
At this point, you should understand the structure and components of the given expression. No further simplification is generally possible unless specific values for [tex]\(x\)[/tex] are provided or specific conditions are given.
Therefore, the expression is:
[tex]\[
\frac{\sin(2x)}{x + \cosh(2x)}
\][/tex]
And this is the detailed breakdown of the expression asked in the question.
