What is the value of -csch^2(2)tanh(2) ?

The value of -csch^2(2)tanh(2) is -(4e^8-4e^4)/(e^{12)-e^8-e^4+1}

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

-csch^2(2)tanh(2)

Solution

- csch^{2} (2) tanh (2)

Solution

- \frac{4 e ^{8} - 4 e ^{4}}{e ^{12} - e ^{8} - e ^{4} + 1}

Decimal Notation

- 0.07328 \dots

Solution steps

- csch^{2} (2) tanh (2)

Rewrite using trig identities:

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

csch (2)

Use the Hyperbolic identity:

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

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

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

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

Apply exponent rule:

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

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

Join

e^{2} - \frac{1}{e ^{2}} : \frac{e ^{4} - 1}{e ^{2}}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

e^{2} e^{2} - 1

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

e^{2} e^{2}

Apply exponent rule:

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

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

= e^{2 + 2}

Add the numbers:

2 + 2 = 4

= e^{4}

= e^{4} - 1

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

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

Apply the fraction rule:

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

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

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

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

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

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

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

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

Apply exponent rule:

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

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

Apply exponent rule:

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

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

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

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

Apply exponent rule:

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

= e^{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= e^{4}

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

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

Multiply fractions:

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

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

2^{2} = 4

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

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

Rewrite using trig identities:

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

tanh (2)

Use the Hyperbolic identity:

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

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

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

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

Apply exponent rule:

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

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

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

Join

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

e^{2} e^{2} + 1

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

e^{2} e^{2}

Apply exponent rule:

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

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

= e^{2 + 2}

Add the numbers:

2 + 2 = 4

= e^{4}

= e^{4} + 1

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

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

Join

e^{2} - \frac{1}{e ^{2}} : \frac{e ^{4} - 1}{e ^{2}}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

e^{2} e^{2} - 1

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

e^{2} e^{2}

Apply exponent rule:

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

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

= e^{2 + 2}

Add the numbers:

2 + 2 = 4

= e^{4}

= e^{4} - 1

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

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

Divide fractions:

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

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

Cancel the common factor:

e^{2}

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

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

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

Simplify

- \frac{4 e ^{4} \frac{e ^{4} - 1}{e ^{4} + 1}}{( e ^{4} - 1 ) ^{2}} : - \frac{4 e ^{8} - 4 e ^{4}}{e ^{12} - e ^{8} - e ^{4} + 1}

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

Multiply

4 e^{4} \frac{e ^{4} - 1}{e ^{4} + 1} : \frac{4 e ^{8} - 4 e ^{4}}{e ^{4} + 1}

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

Multiply fractions:

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

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

Expand

(e^{4} - 1) \cdot 4 e^{4} : 4 e^{8} - 4 e^{4}

(e^{4} - 1) \cdot 4 e^{4}

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

Apply the distributive law:

a (b - c) = ab - a c

a = 4 e^{4}, b = e^{4}, c = 1

= 4 e^{4} e^{4} - 4 e^{4} \cdot 1

= 4 e^{4} e^{4} - 4 \cdot 1 \cdot e^{4}

Simplify

4 e^{4} e^{4} - 4 \cdot 1 \cdot e^{4} : 4 e^{8} - 4 e^{4}

4 e^{4} e^{4} - 4 \cdot 1 \cdot e^{4}

4 e^{4} e^{4} = 4 e^{8}

4 e^{4} e^{4}

Apply exponent rule:

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

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

= 4 e^{4 + 4}

Add the numbers:

4 + 4 = 8

= 4 e^{8}

4 \cdot 1 \cdot e^{4} = 4 e^{4}

4 \cdot 1 \cdot e^{4}

Multiply the numbers:

4 \cdot 1 = 4

= 4 e^{4}

= 4 e^{8} - 4 e^{4}

= 4 e^{8} - 4 e^{4}

= \frac{4 e ^{8} - 4 e ^{4}}{e ^{4} + 1}

= - \frac{\frac{4 e ^{8} - 4 e ^{4}}{e ^{4} + 1}}{( e ^{4} - 1 ) ^{2}}

Apply the fraction rule:

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

= - \frac{4 e ^{8} - 4 e ^{4}}{( e ^{4} + 1 ) ( e ^{4} - 1 ) ^{2}}

Expand

(e^{4} + 1) (e^{4} - 1)^{2} : e^{12} - e^{8} - e^{4} + 1

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

(e^{4} - 1)^{2} : e^{8} - 2 e^{4} + 1

Apply Perfect Square Formula:

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

a = e^{4}, b = 1

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

Simplify

(e^{4})^{2} - 2 e^{4} \cdot 1 + 1^{2} : e^{8} - 2 e^{4} + 1

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

Apply rule

1^{a} = 1

1^{2} = 1

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

(e^{4})^{2} = e^{8}

(e^{4})^{2}

Apply exponent rule:

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

= e^{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= e^{8}

2 e^{4} \cdot 1 = 2 e^{4}

2 e^{4} \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 2 e^{4}

= e^{8} - 2 e^{4} + 1

= e^{8} - 2 e^{4} + 1

= (e^{4} + 1) (e^{8} - 2 e^{4} + 1)

Expand

(e^{4} + 1) (e^{8} - 2 e^{4} + 1) : e^{12} - e^{8} - e^{4} + 1

(e^{4} + 1) (e^{8} - 2 e^{4} + 1)

Distribute parentheses

= e^{4} e^{8} + e^{4} (- 2 e^{4}) + e^{4} \cdot 1 + 1 \cdot e^{8} + 1 \cdot (- 2 e^{4}) + 1 \cdot 1

Apply minus-plus rules

+ (- a) = - a

= e^{8} e^{4} - 2 e^{4} e^{4} + 1 \cdot e^{4} + 1 \cdot e^{8} - 1 \cdot 2 e^{4} + 1 \cdot 1

Simplify

e^{8} e^{4} - 2 e^{4} e^{4} + 1 \cdot e^{4} + 1 \cdot e^{8} - 1 \cdot 2 e^{4} + 1 \cdot 1 : e^{12} - e^{8} - e^{4} + 1

e^{8} e^{4} - 2 e^{4} e^{4} + 1 \cdot e^{4} + 1 \cdot e^{8} - 1 \cdot 2 e^{4} + 1 \cdot 1

e^{8} e^{4} = e^{12}

e^{8} e^{4}

Apply exponent rule:

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

e^{8} e^{4} = e^{8 + 4}

= e^{8 + 4}

Add the numbers:

8 + 4 = 12

= e^{12}

2 e^{4} e^{4} = 2 e^{8}

2 e^{4} e^{4}

Apply exponent rule:

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

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

= 2 e^{4 + 4}

Add the numbers:

4 + 4 = 8

= 2 e^{8}

1 \cdot e^{4} = e^{4}

1 \cdot e^{4}

Multiply:

1 \cdot e^{4} = e^{4}

= e^{4}

1 \cdot e^{8} = e^{8}

1 \cdot e^{8}

Multiply:

1 \cdot e^{8} = e^{8}

= e^{8}

1 \cdot 2 e^{4} = 2 e^{4}

1 \cdot 2 e^{4}

Multiply the numbers:

1 \cdot 2 = 2

= 2 e^{4}

1 \cdot 1 = 1

1 \cdot 1

Multiply the numbers:

1 \cdot 1 = 1

= 1

= e^{12} - 2 e^{8} + e^{4} + e^{8} - 2 e^{4} + 1

Group like terms

= e^{12} - 2 e^{8} + e^{8} + e^{4} - 2 e^{4} + 1

Add similar elements:

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

= e^{12} - 2 e^{8} + e^{8} - e^{4} + 1

Add similar elements:

- 2 e^{8} + e^{8} = - e^{8}

= e^{12} - e^{8} - e^{4} + 1

= e^{12} - e^{8} - e^{4} + 1

= e^{12} - e^{8} - e^{4} + 1

= - \frac{4 e ^{8} - 4 e ^{4}}{e ^{12} - e ^{8} - e ^{4} + 1}

= - \frac{4 e ^{8} - 4 e ^{4}}{e ^{12} - e ^{8} - e ^{4} + 1}

Popular Examples

5*tan(78)

5 \cdot tan (7 8^{\circ})

tan^2(3)

tan^{2} (3)

5/(tan(45))

\frac{5}{tan ( 4 5 ^{\circ} )}

3tan(pi/6)

3 tan (\frac{π}{6})

(4sin(150))/6

\frac{4 sin ( 15 0 ^{\circ} )}{6}

Frequently Asked Questions (FAQ)

What is the value of -csch^2(2)tanh(2) ?
The value of -csch^2(2)tanh(2) is -(4e^8-4e^4)/(e^{12)-e^8-e^4+1}