What is the value of 2-2cosh(1-2) ?

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

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2-2cosh(1-2)

2 - 2 cosh (1 - 2)

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

Decimal Notation

- 1.08616 \dots

Solution steps

2 - 2 cosh (1 - 2)

Simplify

= 2 - 2 cosh (- 1)

Use the following property:

cosh (- x) = cosh (x)

cosh (- 1) = cosh (1)

= 2 - 2 cosh (1)

Rewrite using trig identities:

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

cosh (1)

Use the Hyperbolic identity:

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

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

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

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

Apply rule

a^{1} = a

e^{1} = e

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

Apply exponent rule:

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

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

Join

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

e + \frac{1}{e}

Convert element to fraction:

e = \frac{ee}{e}

= \frac{ee}{e} + \frac{1}{e}

Since the denominators are equal, combine the fractions:

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

= \frac{ee + 1}{e}

ee + 1 = e^{2} + 1

ee + 1

ee = e^{2}

ee

Apply exponent rule:

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

ee = e^{1 + 1}

= e^{1 + 1}

Add the numbers:

1 + 1 = 2

= e^{2}

= e^{2} + 1

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

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

Apply the fraction rule:

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

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

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

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

Simplify

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

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

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

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

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

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