What is the general solution for -cosh(sech(x))(sech(x)tanh(x))=0 ?

The general solution for -cosh(sech(x))(sech(x)tanh(x))=0 is x=0

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-cosh(sech(x))(sech(x)tanh(x))=0

Solution

- cosh (sech (x)) (sech (x) tanh (x)) = 0

Solution

x = 0

Degrees

x = 0^{\circ}

Solution steps

- cosh (sech (x)) (sech (x) tanh (x)) = 0

Rewrite using trig identities

- cosh (sech (x)) sech (x) tanh (x) = 0

Use the Hyperbolic identity:

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

- \frac{e ^{sech (x)} + e ^{- sech (x)}}{2} sech (x) tanh (x) = 0

Use the Hyperbolic identity:

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

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

Use the Hyperbolic identity:

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

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

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

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

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

Multiply both sides by

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

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

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

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

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

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

Apply exponent rule:

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

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

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

Add the numbers:

1 + 1 = 2

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

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

Cancel the common factor:

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

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

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

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

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

Solve

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

No Solution for

x \in R

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

Subtract

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

from both sides

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

Simplify

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

Apply exponent rules

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

f (x) = g (x)

, then

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

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

Apply log rule:

lo g_{c} (ab) = lo g_{c} (a) + lo g_{c} (b)

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

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

Apply log rule:

ln (e^{a}) = a

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

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

Apply log rule:

ln (e^{a}) = a

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

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

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

Multiply both sides by

e^{x} + e^{- x}

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

Simplify

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

Undefined

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

lo g_{a} (b), b < 0

is undefined

= Undefined

2 = Undefined

Therefore

No Solution for x \in R

Solve

e^{x} - e^{- x} = 0 : x = 0

e^{x} - e^{- x} = 0

Add

e^{- x}

to both sides

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

Simplify

e^{x} = e^{- x}

Apply exponent rules

e^{x} = e^{- x}

a^{f (x)} = a^{g (x)}

, then

f (x) = g (x)

x = - x

x = - x

Solve

x = - x : x = 0

x = - x

Move

x

to the left side

x = - x

Add

x

to both sides

x + x = - x + x

Simplify

2 x = 0

2 x = 0

Divide both sides by

2

2 x = 0

Divide both sides by

2

\frac{2 x}{2} = \frac{0}{2}

Simplify

x = 0

x = 0

x = 0

Verify Solutions:

x = 0

True

Check the solutions by plugging them into

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

Remove the ones that don't agree with the equation.

Plug in

x = 0 :

True

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

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

- \frac{e ^{\frac{2}{e ^{0} + e ^{- 0}}} + e ^{- \frac{2}{e ^{0} + e ^{- 0}}}}{2} \cdot \frac{2}{e ^{0} + e ^{- 0}} \cdot \frac{e ^{0} - e ^{- 0}}{e ^{0} + e ^{- 0}}

Apply rule

a^{0} = 1, a \neq = 0

e^{0} = 1, e^{- 0} = 1

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

Multiply fractions:

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

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

Cancel the common factor:

2

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

(e^{\frac{2}{1 + 1}} + e^{- \frac{2}{1 + 1}}) (1 - 1) = 0

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

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

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

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

\frac{2}{1 + 1}

Add the numbers:

1 + 1 = 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

= e^{1}

Apply rule

a^{1} = a

= e

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

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

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

\frac{2}{1 + 1}

Add the numbers:

1 + 1 = 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

= e^{- 1}

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

Subtract the numbers:

1 - 1 = 0

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

Apply rule

0 \cdot a = 0

= 0

= - \frac{0}{( 1 + 1 ) ( 1 + 1 )}

(1 + 1) (1 + 1) = 2^{2}

(1 + 1) (1 + 1)

Apply exponent rule:

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

(1 + 1) (1 + 1) = (1 + 1)^{1 + 1}

= (1 + 1)^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

= - \frac{0}{2 ^{2}}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= - 0

= 0

0 = 0

True

The solution is

x = 0

x = 0

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Frequently Asked Questions (FAQ)

What is the general solution for -cosh(sech(x))(sech(x)tanh(x))=0 ?
The general solution for -cosh(sech(x))(sech(x)tanh(x))=0 is x=0