`sinh(2x) = 2sinhxcoshx` Verify the identity.

Sunday, February 14, 2016

$\displaystyle{\sinh{{\left({2}{x}\right)}}}={2}{\sinh{{x}}}{\cosh{{x}}}$ Verify the identity.

$\displaystyle{\sinh{{\left({2}{x}\right)}}}={2}{\sinh{{\left({x}\right)}}}{\cosh{{\left({x}\right)}}}$

Take note that hyperbolic sine and hyperbolic cosine are defined as

$\displaystyle{\sinh{{\left({u}\right)}}}=\frac{{{{e}}^{{u}}-{{e}}^{{-{u}}}}}{{2}}$

$\displaystyle{\cosh{{\left({u}\right)}}}=\frac{{{{e}}^{{u}}+{{e}}^{{-{u}}}}}{{2}}$

So when the left side is expressed in exponential form, it becomes

$\displaystyle\frac{{{{e}}^{{{2}{x}}}-{{e}}^{{-{2}{x}}}}}{{2}}={2}{\sinh{{\left({x}\right)}}}{\cosh{{\left({x}\right)}}}$

Factoring the numerator, it turns into

$\displaystyle\frac{{{\left({{e}}^{{x}}-{{e}}^{{-{x}}}\right)}{\left({{e}}^{{x}}+{{e}}^{{-{x}}}\right)}}}{{2}}={2}{\sinh{{\left({x}\right)}}}{\cosh{{\left({x}\right)}}}$

To return it to hyperbolic function, multiply the left side by 2/2.

$\displaystyle\frac{{{\left({{e}}^{{x}}-{{e}}^{{-{x}}}\right)}{\left({{e}}^{{x}}+{{e}}^{{-{x}}}\right)}}}{{2}}\cdot\frac{{2}}{{2}}={2}{\sinh{{\left({x}\right)}}}{\cosh{{\left({x}\right)}}}$

Then, rearrange the factors in such a way that it can be expressed in terms of sinh and cosh.

$\displaystyle{2}\cdot\frac{{{{e}}^{{x}}-{{e}}^{{-{x}}}}}{{2}}\cdot\frac{{{{e}}^{{x}}+{{e}}^{{-{x}}}}}{{2}}={2}{\sinh{{\left({x}\right)}}}{\cosh{{\left({x}\right)}}}$

$\displaystyle{2}\cdot{\sinh{{\left({x}\right)}}}\cdot{\cosh{{\left({x}\right)}}}={2}{\sinh{{\left({x}\right)}}}{\cosh{{\left({x}\right)}}}$

$\displaystyle{2}{\sinh{{\left({x}\right)}}}{\cosh{{\left({x}\right)}}}={2}{\sinh{{\left({x}\right)}}}{\cosh{{\left({x}\right)}}}$

This proves that the given equation is an identity.

Therefore, $\displaystyle{\sinh{{\left({2}{x}\right)}}}={2}{\sinh{{\left({x}\right)}}}{\cosh{{\left({x}\right)}}}$ is an identity.

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Sunday, February 14, 2016

$\displaystyle{\sinh{{\left({2}{x}\right)}}}={2}{\sinh{{x}}}{\cosh{{x}}}$ Verify the identity.

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Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

Sunday, February 14, 2016

Verify the identity.

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Post a Comment

Thomas Jefferson&#39;s election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

$\displaystyle{\sinh{{\left({2}{x}\right)}}}={2}{\sinh{{x}}}{\cosh{{x}}}$ Verify the identity.

Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?