`tanh^2(x)+sec^2(x) = 1` Verify the identity.

Friday, January 11, 2013

$\displaystyle{{\tanh}}^{{2}}{\left({x}\right)}+{{\sec}}^{{2}}{\left({x}\right)}={1}$ Verify the identity.

Let's recall the definitions: $\displaystyle{\tanh{{\left({x}\right)}}}=\frac{{{\sinh{{\left({x}\right)}}}}}{{{\cosh{{\left({x}\right)}}}}},$ $\displaystyle{s}{e}{c}{h}{\left({x}\right)}=\frac{{1}}{{{\cosh{{\left({x}\right)}}}}}.$ Also, $\displaystyle{\cosh{{\left({x}\right)}}}=\frac{{{{e}}^{{x}}+{{e}}^{{-{x}}}}}{{2}}$ and $\displaystyle{\sinh{{\left({x}\right)}}}=\frac{{{{e}}^{{x}}-{{e}}^{{-{x}}}}}{{2}}.$

Now the left side of our identity may be rewritten as

$\displaystyle{{\tanh}}^{{2}}{\left({x}\right)}+{{\sec{{h}}}}^{{2}}{\left({x}\right)}=\frac{{{{\sinh}}^{{2}}{\left({x}\right)}+{1}}}{{{{\cosh}}^{{2}}{\left({x}\right)}}}.$

While it is well-known that $\displaystyle{{\sinh}}^{{2}}{\left({x}\right)}+{1}={{\cosh}}^{{2}}{\left({x}\right)},$ we can prove this directly:

$\displaystyle{{\sinh}}^{{2}}{\left({x}\right)}+{1}={{\left(\frac{{{{e}}^{{x}}-{{e}}^{{-{x}}}}}{{2}}\right)}}^{{2}}+{1}=\frac{{{{\left({{e}}^{{x}}\right)}}^{{2}}-{2}+{{\left({{e}}^{{-{x}}}\right)}}^{{2}}+{4}}}{{4}}=$

$\displaystyle=\frac{{{{\left({{e}}^{{x}}\right)}}^{{2}}+{2}+{{\left({{e}}^{{-{x}}}\right)}}^{{2}}}}{{4}}={{\left(\frac{{{{e}}^{{x}}+{{e}}^{{-{x}}}}}{{2}}\right)}}^{{2}}={{\cosh}}^{{2}}{\left({x}\right)}.$

This way the left side is equal to $\displaystyle{1},$ which is the right side. This way the identity is proved.

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Friday, January 11, 2013

$\displaystyle{{\tanh}}^{{2}}{\left({x}\right)}+{{\sec}}^{{2}}{\left({x}\right)}={1}$ 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?

Friday, January 11, 2013

Verify the identity.

No comments:

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?

$\displaystyle{{\tanh}}^{{2}}{\left({x}\right)}+{{\sec}}^{{2}}{\left({x}\right)}={1}$ Verify the identity.

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