What is the general solution for -(tanh(nx))/(cosh(nx))=0 ?

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

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-(tanh(nx))/(cosh(nx))=0

Solution

- \frac{tanh ( n x )}{cosh ( n x )} = 0

Solution

x = 0

Degrees

x = 0^{\circ}

Solution steps

- \frac{tanh ( n x )}{cosh ( n x )} = 0

Rewrite using trig identities

- \frac{tanh ( n x )}{cosh ( n x )} = 0

Use the Hyperbolic identity:

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

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

Use the Hyperbolic identity:

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

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

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

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

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

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

Multiply both sides by

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

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

Simplify

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

Add

e^{- n x}

to both sides

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

Simplify

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

Apply exponent rules

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

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

, then

f (x) = g (x)

n x = - n x

n x = - n x

Solve

n x = - n x : x = 0

n x = - n x

Move

n x

to the left side

n x = - n x

Add

n x

to both sides

n x + n x = - n x + n x

Simplify

2 n x = 0

2 n x = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

x = 0

x = 0

Verify Solutions:

x = 0

True

Check the solutions by plugging them into

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

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

Plug in

x = 0 :

True

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

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

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

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

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

e^{n \cdot 0} - e^{- n \cdot 0} = 0

e^{n \cdot 0} - e^{- n \cdot 0}

e^{n \cdot 0} = 1

e^{n \cdot 0}

Apply rule

0 \cdot a = 0

= e^{0}

Apply rule

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

= 1

e^{- n \cdot 0} = 1

e^{- n \cdot 0}

Apply rule

0 \cdot a = 0

= e^{0}

Apply rule

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

= 1

= 1 - 1

Subtract the numbers:

1 - 1 = 0

= 0

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

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

e^{n \cdot 0} + e^{- n \cdot 0}

e^{n \cdot 0} = 1

e^{n \cdot 0}

Apply rule

0 \cdot a = 0

= e^{0}

Apply rule

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

= 1

e^{- n \cdot 0} = 1

e^{- n \cdot 0}

Apply rule

0 \cdot a = 0

= e^{0}

Apply rule

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

= 1

= 1 + 1

Add the numbers:

1 + 1 = 2

= 2

= \frac{0}{2}

Apply rule

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

= 0

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

\frac{e ^{n \cdot 0} + e ^{- n \cdot 0}}{2} = 1

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

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

e^{n \cdot 0} + e^{- n \cdot 0}

e^{n \cdot 0} = 1

e^{n \cdot 0}

Apply rule

0 \cdot a = 0

= e^{0}

Apply rule

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

= 1

e^{- n \cdot 0} = 1

e^{- n \cdot 0}

Apply rule

0 \cdot a = 0

= e^{0}

Apply rule

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

= 1

= 1 + 1

Add the numbers:

1 + 1 = 2

= 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

= - \frac{0}{1}

Apply rule

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

= 0

0 = 0

True

The solution is

x = 0

x = 0

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