What is the general solution for sinh(x)=sinh(-x) ?

The general solution for sinh(x)=sinh(-x) is x=0

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sinh(x)=sinh(-x)

Solution

sinh (x) = sinh (- x)

Solution

x = 0

Degrees

x = 0^{\circ}

Solution steps

sinh (x) = sinh (- x)

Rewrite using trig identities

sinh (x) = sinh (- x)

Use the Hyperbolic identity:

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

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

Use the Hyperbolic identity:

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

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

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

\frac{e ^{x} - e ^{- x}}{2} = \frac{e ^{- x} - e ^{- (- x)}}{2} : x = 0

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

Multiply both sides by

2

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

Simplify

e^{x} - e^{- x} = e^{- x} - e^{- (- x)}

Apply exponent rules

e^{x} - e^{- x} = e^{- x} - e^{- (- x)}

Apply exponent rule:

a^{b c} = (a^{b})^{c}

e^{- x} = (e^{x})^{- 1}

e^{x} - (e^{x})^{- 1} = (e^{x})^{- 1} - e^{- (- x)}

e^{x} - (e^{x})^{- 1} = (e^{x})^{- 1} - e^{- (- x)}

Subtract

(e^{x})^{- 1} - e^{- (- x)}

from both sides

e^{x} - (e^{x})^{- 1} - ((e^{x})^{- 1} - e^{- (- x)}) = (e^{x})^{- 1} - e^{- (- x)} - ((e^{x})^{- 1} - e^{- (- x)})

Simplify

e^{x} - (e^{x})^{- 1} - (e^{x})^{- 1} + e^{- (- x)} : e^{x} - \frac{2}{e ^{x}} + e^{- (- x)}

e^{x} - (e^{x})^{- 1} - (e^{x})^{- 1} + e^{- (- x)}

Add similar elements:

= e^{x} - 2 (e^{x})^{- 1} + e^{- (- x)}

2 (e^{x})^{- 1}

Apply exponent rule:

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

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

Simplify

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

(a_{4})^{- 1} - e^{- (- x)} - (e^{x})^{- 1} + e^{- (- x)}

Group like terms

= (e^{- (- x)})^{- 1} - e^{- (- x)} + e^{- (- x)} - (e^{x})^{- 1}

Add similar elements:

- e^{- (- x)} + e^{- (- x)} = 0

= (e^{- (- x)})^{- 1} - (e^{x})^{- 1}

Apply exponent rule:

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

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

Apply exponent rule:

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

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

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

Subtract

\frac{1}{e ^{- (- x)}} - \frac{1}{e ^{x}}

from both sides

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

Simplify

- \frac{1}{e ^{x}} + e^{x} + e^{- (- x)} - \frac{1}{e ^{- (- x)}} = 0

Factor

- \frac{2}{e ^{x}} + 2 e^{x} = 0

Add

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

to both sides

- \frac{2}{e ^{x}} + 2 e^{x} + \frac{2}{e ^{x}} = 0 + \frac{2}{e ^{x}}

Simplify

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

Rewrite the equation with

e^{x} = u

2 u = \frac{2}{u}

Solve

2 u = \frac{2}{u} : u = 1, u = - 1

2 u = \frac{2}{u}

Multiply both sides by

u

2 u = \frac{2}{u}

Multiply both sides by

u

2 uu = \frac{2}{u} u

Simplify

2 uu : 2 u^{2}

2 uu = \frac{2}{u} u

Apply exponent rule:

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

uu = u^{1 + 1}

= 2 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2 u^{2}

2 u^{2} = 2

2 u^{2} = 2

Solve

2 u^{2} = 2 : u = 1, u = - 1

2 u^{2} = 2

Divide both sides by

2

2 u^{2} = 2

Divide both sides by

2

\frac{2 u ^{2}}{2} = \frac{2}{2}

Simplify

u^{2} = 1

u^{2} = 1

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = 1, u = - 1

1 = 1

1

Apply radical rule:

1 = 1

= 1

- 1 = - 1

- 1

Apply radical rule:

1 = 1

1 = 1

= - 1

u = 1, u = - 1

u = 1, u = - 1

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

\frac{2}{u}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 1, u = - 1

u = 1, u = - 1

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = 1 : x = 0

e^{x} = 1

Apply exponent rules

e^{x} = 1

f (x) = g (x)

, then

ln (f (x)) = ln (g (x))

ln (e^{x}) = ln (1)

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (1)

Simplify

ln (1) : 0

ln (1)

Apply log rule:

lo g_{a} (1) = 0

= 0

x = 0

x = 0

Solve

e^{x} = - 1 :

No Solution for

x \in R

e^{x} = - 1

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

x = 0

x = 0

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What is the general solution for sinh(x)=sinh(-x) ?
The general solution for sinh(x)=sinh(-x) is x=0