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zhannawk [14.2K]
2 years ago
5

Please help me answer this question

Mathematics
1 answer:
ratelena [41]2 years ago
6 0

By the definition of the hyperbolic function tanh x, we have proven that \frac{1-tanh \ x}{1 + tanh \ x}=e^{-2x}

<h3>Hyperbolic functions & Proof of identities </h3>

By definition

tanh \ x=\frac{e^{x} -e^{-x}}{e^{x} +e^{-x}}

Then,

\frac{1-tanh \ x}{1 + tanh \ x}=\frac{1-\frac{e^{x} -e^{-x}}{e^{x} +e^{-x}} }{1+ \frac{e^{x} -e^{-x}}{e^{x} +e^{-x}} }

=1-\frac{e^{x} -e^{-x}}{e^{x} +e^{-x}} \div (1+ \frac{e^{x} -e^{-x}}{e^{x} +e^{-x}} })

=\frac{e^{x} +e^{-x}-(e^{x} -e^{-x})}{e^{x} +e^{-x}} \div (\frac{e^{x} +e^{-x}+e^{x} -e^{-x}}{e^{x} +e^{-x}} })

=\frac{e^{x} +e^{-x}-e^{x} +e^{-x}}{e^{x} +e^{-x}} \div \frac{e^{x} +e^{-x}+e^{x} -e^{-x}}{e^{x} +e^{-x}} }

=\frac{e^{-x}+e^{-x}}{e^{x} +e^{-x}} \div \frac{e^{x} +e^{x} }{e^{x} +e^{-x}} }

=\frac{2e^{-x}}{e^{x} +e^{-x}} \div \frac{2e^{x} }{e^{x} +e^{-x}} }

=\frac{2e^{-x}}{e^{x} +e^{-x}} \times \frac{e^{x} +e^{-x}}{2e^{x}}

=\frac{2e^{-x}}{1} \times \frac{1}{2e^{x}}

=\frac{2e^{-x}}{2e^{x}}

=\frac{e^{-x}}{e^{x}}
=e^{-x} \times \frac{1}{e^{x}}

= e^{-x} \times e^{-x}

= e^{-x+-x}

= e^{-x-x}

= e^{-2x}

Hence, we have proven that \frac{1-tanh \ x}{1 + tanh \ x}=e^{-2x}

Learn more on Proof of Identities here: brainly.com/question/2561079

#SPJ1

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