What is the value of 1/2 sech^2(1)-1/2 tanh(1) ?

The value of 1/2 sech^2(1)-1/2 tanh(1) is (2e^2)/((e^2+1)^2)-(e^2-1)/(2(e^2+1))

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

1/2 sech^2(1)-1/2 tanh(1)

Solution

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

Solution

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

Decimal Notation

- 0.17080 \dots

Solution steps

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

Rewrite using trig identities:

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

sech (1)

Use the Hyperbolic identity:

sech (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}

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

Simplify

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

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

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

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

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

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

Apply exponent rule:

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

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

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

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

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

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

Multiply fractions:

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

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

Multiply:

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

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

Cancel the common factor:

2

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

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

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

Multiply fractions:

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

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

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

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

Multiply:

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

= (e^{2} - 1)

Remove parentheses:

(a) = a

= e^{2} - 1

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

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

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

Popular Examples

3cot(45)

3 cot (4 5^{\circ})

0.9144+3.048sin(30)

0.9144 + 3.048 sin (3 0^{\circ})

(3(2)^2-sin(2))/(2e^2)

\frac{3 ( 2 ) ^{2} - sin ( 2 )}{2 e ^{2}}

arctan(9/(-2))

arctan (\frac{9}{- 2})

arctan(-1/8)

arctan (- \frac{1}{8})

Frequently Asked Questions (FAQ)

What is the value of 1/2 sech^2(1)-1/2 tanh(1) ?
The value of 1/2 sech^2(1)-1/2 tanh(1) is (2e^2)/((e^2+1)^2)-(e^2-1)/(2(e^2+1))