What is the value of csch(pi) ?

The value of csch(pi) is (2e^pi)/(e^{2pi)-1}

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csch(pi)

csch (π)

\frac{2 e ^{π}}{e ^{2 π} - 1}

Decimal

0.08658 \dots

Solution steps

csch (π)

Use the Hyperbolic identity:

csch (x) = \frac{2}{e ^{x} - e ^{- x}}

= \frac{2}{e ^{π} - e ^{- π}}

\frac{2}{e ^{π} - e ^{- π}} = \frac{2 e ^{π}}{e ^{2 π} - 1}

\frac{2}{e ^{π} - e ^{- π}}

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

= \frac{2}{e ^{π} - \frac{1}{e ^{π}}}

Join

e^{π} - \frac{1}{e ^{π}} : \frac{e ^{2 π} - 1}{e ^{π}}

e^{π} - \frac{1}{e ^{π}}

Convert element to fraction:

e^{π} = \frac{e ^{π} e ^{π}}{e ^{π}}

= \frac{e ^{π} e ^{π}}{e ^{π}} - \frac{1}{e ^{π}}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{e ^{π} e ^{π} - 1}{e ^{π}}

e^{π} e^{π} - 1 = e^{2 π} - 1

e^{π} e^{π} - 1

e^{π} e^{π} = e^{2 π}

e^{π} e^{π}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

e^{π} e^{π} = e^{π + π}

= e^{π + π}

Add similar elements:

π + π = 2 π

= e^{2 π}

= e^{2 π} - 1

= \frac{e ^{2 π} - 1}{e ^{π}}

= \frac{2}{\frac{e ^{2 π} - 1}{e ^{π}}}

Apply the fraction rule:

\frac{a}{\frac{b}{c}} = \frac{a \cdot c}{b}

= \frac{2 e ^{π}}{e ^{2 π} - 1}

= \frac{2 e ^{π}}{e ^{2 π} - 1}