What is the value of 2-2cosh(-2) ?

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

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2-2cosh(-2)

2 - 2 cosh (- 2)

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

Decimal Notation

- 5.52439 \dots

Solution steps

2 - 2 cosh (- 2)

Rewrite using trig identities:

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

cosh (- 2)

Use the following property:

cosh (- x) = cosh (x)

cosh (- 2) = cosh (2)

= cosh (2)

Rewrite using trig identities:

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

cosh (2)

Use the Hyperbolic identity:

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

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

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

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

Apply exponent rule:

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

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

Apply the fraction rule:

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

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

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

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

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

Simplify

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

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

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

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

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

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