Is ((sech^2(x)))/(tanh(x))=csch(x) ?

The answer to whether ((sech^2(x)))/(tanh(x))=csch(x) is False

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prove ((sech^2(x)))/(tanh(x))=csch(x)

Solution

prove

\frac{( sech ^{2} ( x ) )}{tanh ( x )} = csch (x)

Solution

False

Solution steps

\frac{sech ^{2} ( x )}{tanh ( x )} = csch (x)

Show that the two sides are not equal

For

\frac{sech ^{2} ( x )}{tanh ( x )} = csch (x)

plug in

x = 1

\frac{sech ^{2} ( 1 )}{tanh ( 1 )} = \frac{4 e ^{2}}{( e ^{2} + 1 ) ( e ^{2} - 1 )} (Decimal : 0.55144 \dots)

\frac{sech ^{2} ( 1 )}{tanh ( 1 )}

Rewrite using trig identities:

sech^{2} (1) = 1 - tanh^{2} (1)

sech^{2} (1)

Use the Hyperbolic identity:

sech^{2} (x) = 1 - tanh^{2} (x)

= 1 - tanh^{2} (1)

= \frac{1 - tanh ^{2} ( 1 )}{tanh ( 1 )}

Rewrite using trig identities:

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

tanh (1)

Use the Hyperbolic identity:

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

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

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

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

Apply rule

a^{1} = a

e^{1} = e

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

Apply exponent rule:

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

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

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

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

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

Divide fractions:

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

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

Cancel the common factor:

e

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

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

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

Simplify

\frac{1 - ( \frac{e ^{2} - 1}{e ^{2} + 1} ) ^{2}}{\frac{e ^{2} - 1}{e ^{2} + 1}} : \frac{4 e ^{2}}{( e ^{2} + 1 ) ( e ^{2} - 1 )}

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

Apply the fraction rule:

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

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

Apply exponent rule:

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

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

Join

1 - \frac{( e ^{2} - 1 ) ^{2}}{( e ^{2} + 1 ) ^{2}} : \frac{4 e ^{2}}{( e ^{2} + 1 ) ^{2}}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

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

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

Expand

(e^{2} + 1)^{2} - (e^{2} - 1)^{2} : 4 e^{2}

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

(e^{2} + 1)^{2} : e^{4} + 2 e^{2} + 1

Apply Perfect Square Formula:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = e^{2}, b = 1

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

Simplify

(e^{2})^{2} + 2 e^{2} \cdot 1 + 1^{2} : e^{4} + 2 e^{2} + 1

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

Apply rule

1^{a} = 1

1^{2} = 1

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

(e^{2})^{2} = e^{4}

(e^{2})^{2}

Apply exponent rule:

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

= e^{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= e^{4}

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

2 e^{2} \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 2 e^{2}

= e^{4} + 2 e^{2} + 1

= e^{4} + 2 e^{2} + 1

= e^{4} + 2 e^{2} + 1 - (e^{2} - 1)^{2}

(e^{2} - 1)^{2} : e^{4} - 2 e^{2} + 1

Apply Perfect Square Formula:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = e^{2}, b = 1

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

Simplify

(e^{2})^{2} - 2 e^{2} \cdot 1 + 1^{2} : e^{4} - 2 e^{2} + 1

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

Apply rule

1^{a} = 1

1^{2} = 1

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

(e^{2})^{2} = e^{4}

(e^{2})^{2}

Apply exponent rule:

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

= e^{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= e^{4}

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

2 e^{2} \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 2 e^{2}

= e^{4} - 2 e^{2} + 1

= e^{4} - 2 e^{2} + 1

= e^{4} + 2 e^{2} + 1 - (e^{4} - 2 e^{2} + 1)

- (e^{4} - 2 e^{2} + 1) : - e^{4} + 2 e^{2} - 1

- (e^{4} - 2 e^{2} + 1)

Distribute parentheses

= - (e^{4}) - (- 2 e^{2}) - (1)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - e^{4} + 2 e^{2} - 1

= e^{4} + 2 e^{2} + 1 - e^{4} + 2 e^{2} - 1

Simplify

e^{4} + 2 e^{2} + 1 - e^{4} + 2 e^{2} - 1 : 4 e^{2}

e^{4} + 2 e^{2} + 1 - e^{4} + 2 e^{2} - 1

Group like terms

= e^{4} - e^{4} + 2 e^{2} + 2 e^{2} + 1 - 1

Add similar elements:

2 e^{2} + 2 e^{2} = 4 e^{2}

= e^{4} - e^{4} + 4 e^{2} + 1 - 1

Add similar elements:

e^{4} - e^{4} = 0

= 4 e^{2} + 1 - 1

1 - 1 = 0

= 4 e^{2}

= 4 e^{2}

= \frac{4 e ^{2}}{( e ^{2} + 1 ) ^{2}}

= \frac{\frac{4 e ^{2}}{( e ^{2} + 1 ) ^{2}} ( e ^{2} + 1 )}{e ^{2} - 1}

Multiply

\frac{4 e ^{2}}{( e ^{2} + 1 ) ^{2}} (e^{2} + 1) : \frac{4 e ^{2}}{e ^{2} + 1}

\frac{4 e ^{2}}{( e ^{2} + 1 ) ^{2}} (e^{2} + 1)

Multiply fractions:

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

= \frac{4 e ^{2} ( e ^{2} + 1 )}{( e ^{2} + 1 ) ^{2}}

Cancel the common factor:

e^{2} + 1

= \frac{4 e ^{2}}{e ^{2} + 1}

= \frac{\frac{4 e ^{2}}{e ^{2} + 1}}{e ^{2} - 1}

Apply the fraction rule:

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

= \frac{4 e ^{2}}{( e ^{2} + 1 ) ( e ^{2} - 1 )}

= \frac{4 e ^{2}}{( e ^{2} + 1 ) ( e ^{2} - 1 )}

csch (1) = \frac{2 e}{e ^{2} - 1} (Decimal : 0.85091 \dots)

csch (1)

Use the Hyperbolic identity:

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

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

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

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

Apply rule

a^{1} = a

e^{1} = e

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

Apply exponent rule:

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

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

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

Apply the fraction rule:

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

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

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

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

The two sides are not equal

\Rightarrow False

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Frequently Asked Questions (FAQ)

Is ((sech^2(x)))/(tanh(x))=csch(x) ?
The answer to whether ((sech^2(x)))/(tanh(x))=csch(x) is False