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

The general solution for (cosh(x))=sinh(x) is No Solution for x\in\mathbb{R}

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(cosh(x))=sinh(x)

Solution

(cosh (x)) = sinh (x)

Solution

No Solution for x \in R

Solution steps

(cosh (x)) = sinh (x)

Rewrite using trig identities

cosh (x) = sinh (x)

Use the Hyperbolic identity:

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

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

Use the Hyperbolic identity:

cosh (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} :

No Solution for

x \in R

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

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

Rewrite the equation with

e^{x} = u

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

Solve

u + u^{- 1} = u - u^{- 1} :

No Solution for

u \in R

u + u^{- 1} = u - u^{- 1}

Refine

u + \frac{1}{u} = u - \frac{1}{u}

Subtract

u

from both sides

u + \frac{1}{u} - u = u - \frac{1}{u} - u

Simplify

\frac{1}{u} = - \frac{1}{u}

Multiply both sides by

u

\frac{1}{u} = - \frac{1}{u}

Multiply both sides by

u

\frac{1}{u} u = - \frac{1}{u} u

Simplify

\frac{1}{u} u = - \frac{1}{u} u

Simplify

\frac{1}{u} u : 1

\frac{1}{u} u

Multiply fractions:

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

= \frac{1 \cdot u}{u}

Cancel the common factor:

u

= 1

Simplify

- \frac{1}{u} u : - 1

- \frac{1}{u} u

Multiply fractions:

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

= - \frac{1 \cdot u}{u}

Cancel the common factor:

u

= - 1

1 = - 1

1 = - 1

1 = - 1

The sides are not equal

No Solution for u \in R

No Solution for x \in R

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for (cosh(x))=sinh(x) ?
The general solution for (cosh(x))=sinh(x) is No Solution for x\in\mathbb{R}