What is the value of coth(2) ?

The value of coth(2) is (e^4+1)/(e^4-1)

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coth(2)

coth (2)

\frac{e ^{4} + 1}{e ^{4} - 1}

Decimal Notation

1.03731 \dots

Solution steps

coth (2)

Use the Hyperbolic identity:

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

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

\frac{e ^{2} + e ^{- 2}}{e ^{2} - e ^{- 2}} = \frac{e ^{4} + 1}{e ^{4} - 1}

\frac{e ^{2} + e ^{- 2}}{e ^{2} - e ^{- 2}}

Apply exponent rule:

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

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

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

Join

e^{2} - \frac{1}{e ^{2}} : \frac{e ^{4} - 1}{e ^{2}}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

e^{2} e^{2} - 1 = e^{4} - 1

e^{2} e^{2} - 1

e^{2} e^{2} = e^{4}

e^{2} e^{2}

Apply exponent rule:

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

e^{2} e^{2} = e^{2 + 2}

= e^{2 + 2}

Add the numbers:

2 + 2 = 4

= e^{4}

= e^{4} - 1

= \frac{e ^{4} - 1}{e ^{2}}

= \frac{e ^{2} + \frac{1}{e ^{2}}}{\frac{e ^{4} - 1}{e ^{2}}}

Join

e^{2} + \frac{1}{e ^{2}} : \frac{e ^{4} + 1}{e ^{2}}

e^{2} + \frac{1}{e ^{2}}

Convert element to fraction:

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

= \frac{e ^{2} e ^{2}}{e ^{2}} + \frac{1}{e ^{2}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{2} e ^{2} + 1}{e ^{2}}

e^{2} e^{2} + 1 = e^{4} + 1

e^{2} e^{2} + 1

e^{2} e^{2} = e^{4}

e^{2} e^{2}

Apply exponent rule:

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

e^{2} e^{2} = e^{2 + 2}

= e^{2 + 2}

Add the numbers:

2 + 2 = 4

= e^{4}

= e^{4} + 1

= \frac{e ^{4} + 1}{e ^{2}}

= \frac{\frac{e ^{4} + 1}{e ^{2}}}{\frac{e ^{4} - 1}{e ^{2}}}

Divide fractions:

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

= \frac{( e ^{4} + 1 ) e ^{2}}{e ^{2} ( e ^{4} - 1 )}

Cancel the common factor:

e^{2}

= \frac{e ^{4} + 1}{e ^{4} - 1}

= \frac{e ^{4} + 1}{e ^{4} - 1}

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