What is the value of (tan((5pi)/8)-tan((3pi)/8))/(1+tan((5pi)/8)tan((3pi)/8)) ?

The value of (tan((5pi)/8)-tan((3pi)/8))/(1+tan((5pi)/8)tan((3pi)/8)) is 1

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(tan((5pi)/8)-tan((3pi)/8))/(1+tan((5pi)/8)tan((3pi)/8))

Solution

\frac{tan ( \frac{5 π}{8} ) - tan ( \frac{3 π}{8} )}{1 + tan ( \frac{5 π}{8} ) tan ( \frac{3 π}{8} )}

Solution

1

Solution steps

\frac{tan ( \frac{5 π}{8} ) - tan ( \frac{3 π}{8} )}{1 + tan ( \frac{5 π}{8} ) tan ( \frac{3 π}{8} )}

\frac{tan ( \frac{5 π}{8} ) - tan ( \frac{3 π}{8} )}{1 + tan ( \frac{5 π}{8} ) tan ( \frac{3 π}{8} )} = \frac{tan ( \frac{3 π}{8} ) - tan ( \frac{5 π}{8} )}{- tan ( \frac{5 π}{8} ) tan ( \frac{3 π}{8} ) - 1}

\frac{tan ( \frac{5 π}{8} ) - tan ( \frac{3 π}{8} )}{1 + tan ( \frac{5 π}{8} ) tan ( \frac{3 π}{8} )}

Apply the fraction rule:

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

\frac{tan ( \frac{5 π}{8} ) - tan ( \frac{3 π}{8} )}{1 + tan ( \frac{5 π}{8} ) tan ( \frac{3 π}{8} )} = \frac{- ( tan ( \frac{3 π}{8} ) - tan ( \frac{5 π}{8} ) )}{- ( - tan ( \frac{5 π}{8} ) tan ( \frac{3 π}{8} ) - 1 )}

= \frac{tan ( \frac{3 π}{8} ) - tan ( \frac{5 π}{8} )}{- tan ( \frac{5 π}{8} ) tan ( \frac{3 π}{8} ) - 1}

= \frac{tan ( \frac{3 π}{8} ) - tan ( \frac{5 π}{8} )}{- tan ( \frac{5 π}{8} ) tan ( \frac{3 π}{8} ) - 1}

Rewrite using trig identities:

tan (\frac{5 π}{8}) = - tan (\frac{3 π}{8})

tan (\frac{5 π}{8})

Use the following identity:

tan (x) = - tan (π - x)

tan (x)

Use the following property:

tan (θ) = - tan (- θ)

tan (x) = - tan (- x)

= - tan (- x)

Apply the periodicity of

tan

tan (π + θ) = tan (θ)

- tan (- x) = - tan (π - x)

= - tan (π - x)

= - tan (2 π - \frac{5 π}{8})

Simplify:

2 π - \frac{5 π}{8} = \frac{11 π}{8}

2 π - \frac{5 π}{8}

Convert element to fraction:

2 π = \frac{2 π 8}{8}

= \frac{2 π 8}{8} - \frac{5 π}{8}

Since the denominators are equal, combine the fractions:

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

= \frac{2 π 8 - 5 π}{8}

2 π 8 - 5 π = 11 π

2 π 8 - 5 π

Multiply the numbers:

2 \cdot 8 = 16

= 16 π - 5 π

Add similar elements:

16 π - 5 π = 11 π

= 11 π

= \frac{11 π}{8}

= - tan (\frac{11 π}{8})

tan (\frac{11 π}{8}) = tan (\frac{3 π}{8})

tan (\frac{11 π}{8})

Rewrite

\frac{11 π}{8}

π + \frac{3 π}{8}

= tan (π + \frac{3 π}{8})

Apply the periodicity of

tan

tan (x + π) = tan (x)

tan (π + \frac{3 π}{8}) = tan (\frac{3 π}{8})

= tan (\frac{3 π}{8})

= - tan (\frac{3 π}{8})

= \frac{tan ( \frac{3 π}{8} ) - ( - tan ( \frac{3 π}{8} ) )}{- ( - tan ( \frac{3 π}{8} ) ) tan ( \frac{3 π}{8} ) - 1}

\frac{tan ( \frac{3 π}{8} ) - ( - tan ( \frac{3 π}{8} ) )}{- ( - tan ( \frac{3 π}{8} ) ) tan ( \frac{3 π}{8} ) - 1} = \frac{2 tan ( \frac{3 π}{8} )}{tan ^{2} ( \frac{3 π}{8} ) - 1}

\frac{tan ( \frac{3 π}{8} ) - ( - tan ( \frac{3 π}{8} ) )}{- ( - tan ( \frac{3 π}{8} ) ) tan ( \frac{3 π}{8} ) - 1}

Apply rule

- (- a) = a

= \frac{tan ( \frac{3 π}{8} ) + tan ( \frac{3 π}{8} )}{tan ( \frac{3 π}{8} ) tan ( \frac{3 π}{8} ) - 1}

Add similar elements:

tan (\frac{3 π}{8}) + tan (\frac{3 π}{8}) = 2 tan (\frac{3 π}{8})

= \frac{2 tan ( \frac{3 π}{8} )}{tan ( \frac{3 π}{8} ) tan ( \frac{3 π}{8} ) - 1}

tan (\frac{3 π}{8}) tan (\frac{3 π}{8}) - 1 = tan^{2} (\frac{3 π}{8}) - 1

tan (\frac{3 π}{8}) tan (\frac{3 π}{8}) - 1

tan (\frac{3 π}{8}) tan (\frac{3 π}{8}) = tan^{2} (\frac{3 π}{8})

tan (\frac{3 π}{8}) tan (\frac{3 π}{8})

Apply exponent rule:

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

tan (\frac{3 π}{8}) tan (\frac{3 π}{8}) = tan^{1 + 1} (\frac{3 π}{8})

= tan^{1 + 1} (\frac{3 π}{8})

Add the numbers:

1 + 1 = 2

= tan^{2} (\frac{3 π}{8})

= tan^{2} (\frac{3 π}{8}) - 1

= \frac{2 tan ( \frac{3 π}{8} )}{tan ^{2} ( \frac{3 π}{8} ) - 1}

= \frac{2 tan ( \frac{3 π}{8} )}{tan ^{2} ( \frac{3 π}{8} ) - 1}

Rewrite using trig identities:

tan (\frac{3 π}{8}) = 3 + 22

tan (\frac{3 π}{8})

Rewrite using trig identities:

\frac{1 - cos ( \frac{3 π}{4} )}{1 + cos ( \frac{3 π}{4} )}

tan (\frac{3 π}{8})

Write

tan (\frac{3 π}{8})

tan (\frac{\frac{3 π}{4}}{2})

= tan (\frac{\frac{3 π}{4}}{2})

Use the Half Angle identity:

tan (\frac{θ}{2}) = \frac{1 - cos ( θ )}{1 + cos ( θ )}

Rewrite using trig identities:

tan^{2} (θ) = \frac{1 - cos ( 2 θ )}{1 + cos ( 2 θ )}

Use the following identity

tan (θ) = \frac{sin ( θ )}{cos ( θ )}

Square both sides

tan^{2} (θ) = \frac{sin ^{2} ( θ )}{cos ^{2} ( θ )}

Rewrite using trig identities:

sin^{2} (θ) = \frac{1 - cos ( 2 θ )}{2}

Use the Double Angle identity

cos (2 θ) = 1 - 2 sin^{2} (θ)

Switch sides

2 sin^{2} (θ) - 1 = - cos (2 θ)

Add

1

to both sides

2 sin^{2} (θ) = 1 - cos (2 θ)

Divide both sides by

2

sin^{2} (θ) = \frac{1 - cos ( 2 θ )}{2}

Rewrite using trig identities:

cos^{2} (θ) = \frac{1 + cos ( 2 θ )}{2}

Use the Double Angle identity

cos (2 θ) = 2 cos^{2} (θ) - 1

Switch sides

2 cos^{2} (θ) - 1 = cos (2 θ)

Add

1

to both sides

2 sin^{2} (θ) = 1 + cos (2 θ)

Divide both sides by

2

cos^{2} (θ) = \frac{1 + cos ( 2 θ )}{2}

tan^{2} (θ) = \frac{\frac{1 - c o s ( 2 θ )}{2}}{\frac{1 + c o s ( 2 θ )}{2}}

Simplify

tan^{2} (θ) = \frac{1 - cos ( 2 θ )}{1 + cos ( 2 θ )}

Substitute

θ

with

\frac{θ}{2}

tan^{2} (\frac{θ}{2}) = \frac{1 - cos ( 2 \cdot \frac{θ}{2} )}{1 + cos ( 2 \cdot \frac{θ}{2} )}

Simplify

tan^{2} (\frac{θ}{2}) = \frac{1 - cos ( θ )}{1 + cos ( θ )}

Square root both sides

Choose the root sign according to the quadrant of

\frac{θ}{2}

range [0, \frac{π}{2}] [\frac{π}{2}, π] quadrant I II tan positive negative

tan (\frac{θ}{2}) = \frac{1 - cos ( θ )}{1 + cos ( θ )}

= \frac{1 - cos ( \frac{3 π}{4} )}{1 + cos ( \frac{3 π}{4} )}

= \frac{1 - cos ( \frac{3 π}{4} )}{1 + cos ( \frac{3 π}{4} )}

Use the following trivial identity:

cos (\frac{3 π}{4}) = - \frac{2}{2}

cos (\frac{3 π}{4})

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= - \frac{2}{2}

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

Simplify

\frac{1 - ( - \frac{2}{2} )}{1 - \frac{2}{2}} : 3 + 22

\frac{1 - ( - \frac{2}{2} )}{1 - \frac{2}{2}}

Apply rule

- (- a) = a

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

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

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

Join

1 - \frac{2}{2} : \frac{2 - 2}{2}

1 - \frac{2}{2}

Convert element to fraction:

1 = \frac{1 \cdot 2}{2}

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

Since the denominators are equal, combine the fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2 - 2}{2}

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

Join

1 + \frac{2}{2} : \frac{2 + 2}{2}

1 + \frac{2}{2}

Convert element to fraction:

1 = \frac{1 \cdot 2}{2}

= \frac{1 \cdot 2}{2} + \frac{2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 2 + 2}{2}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2 + 2}{2}

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

Divide fractions:

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

= \frac{( 2 + 2 ) \cdot 2}{2 ( 2 - 2 )}

Cancel the common factor:

2

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

Factor

2 + 2 : 2 (2 + 1)

2 + 2

2 = 22

= 22 + 2

Factor out common term

2

= 2 (2 + 1)

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

Factor

2 - 2 : 2 (2 - 1)

2 - 2

2 = 22

= 22 - 2

Factor out common term

2

= 2 (2 - 1)

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

Cancel the common factor:

2

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

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

\frac{2 + 1}{2 - 1} = 3 + 22

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

Multiply by the conjugate

\frac{2 + 1}{2 + 1}

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

(2 + 1) (2 + 1) = 3 + 22

(2 + 1) (2 + 1)

Apply exponent rule:

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

(2 + 1) (2 + 1) = (2 + 1)^{1 + 1}

= (2 + 1)^{1 + 1}

Add the numbers:

1 + 1 = 2

= (2 + 1)^{2}

Apply Perfect Square Formula:

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

a = 2, b = 1

= (2)^{2} + 22 \cdot 1 + 1^{2}

Simplify

(2)^{2} + 22 \cdot 1 + 1^{2} : 3 + 22

(2)^{2} + 22 \cdot 1 + 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

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

(2)^{2} = 2

(2)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

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

Apply exponent rule:

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

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

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2

22 \cdot 1 = 22

22 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 22

= 2 + 22 + 1

Add the numbers:

2 + 1 = 3

= 3 + 22

= 3 + 22

(2 - 1) (2 + 1) = 1

(2 - 1) (2 + 1)

Apply Difference of Two Squares Formula:

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

a = 2, b = 1

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

Simplify

(2)^{2} - 1^{2} : 1

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

Apply rule

1^{a} = 1

1^{2} = 1

= (2)^{2} - 1

(2)^{2} = 2

(2)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

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

Apply exponent rule:

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

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

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2

= 2 - 1

Subtract the numbers:

2 - 1 = 1

= 1

= 1

= \frac{3 + 2 2}{1}

Apply rule

\frac{a}{1} = a

= 3 + 22

= 3 + 22

= 3 + 22

= \frac{2 3 + 2 2}{( 3 + 2 2 ) ^{2} - 1}

Simplify

\frac{2 3 + 2 2}{( 3 + 2 2 ) ^{2} - 1} : 1

\frac{2 3 + 2 2}{( 3 + 2 2 ) ^{2} - 1}

Factor

(3 + 22)^{2} - 1 : 2 (2 + 2)

(3 + 22)^{2} - 1

Rewrite

1

1^{2}

= (3 + 22)^{2} - 1^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

(3 + 22)^{2} - 1^{2} = (3 + 22 + 1) (3 + 22 - 1)

= (3 + 22 + 1) (3 + 22 - 1)

(3 + 22 + 1) (3 + 22 - 1) = 2 (2 + 2)

(3 + 22 + 1) (3 + 22 - 1)

3 + 22 = 2 + 1

3 + 22

= 2 + 22 + 1

= (2)^{2} + 22 + (1)^{2}

1 = 1

1

Apply rule

1 = 1

= 1

= (2)^{2} + 22 + 1^{2}

22 \cdot 1 = 22

22 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 22

= (2)^{2} + 22 \cdot 1 + 1^{2}

Apply Perfect Square Formula:

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

(2)^{2} + 22 \cdot 1 + 1^{2} = (2 + 1)^{2}

= (2 + 1)^{2}

Apply radical rule:

(2 + 1)^{2} = 2 + 1

= 2 + 1

= (1 + 1 + 2) (3 + 22 - 1)

3 + 22 = 2 + 1

3 + 22

= 2 + 22 + 1

= (2)^{2} + 22 + (1)^{2}

1 = 1

1

Apply rule

1 = 1

= 1

= (2)^{2} + 22 + 1^{2}

22 \cdot 1 = 22

22 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 22

= (2)^{2} + 22 \cdot 1 + 1^{2}

Apply Perfect Square Formula:

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

(2)^{2} + 22 \cdot 1 + 1^{2} = (2 + 1)^{2}

= (2 + 1)^{2}

Apply radical rule:

(2 + 1)^{2} = 2 + 1

= 2 + 1

= (1 + 1 + 2) (1 + 2 - 1)

Add the numbers:

1 + 1 = 2

= (2 + 2) (1 + 2 - 1)

1 - 1 = 0

= 2 (2 + 2)

= 2 (2 + 2)

= \frac{2 3 + 2 2}{2 ( 2 + 2 )}

Cancel

\frac{2 3 + 2 2}{2 ( 2 + 2 )} : \frac{2 3 + 2 2}{2 + 2}

\frac{2 3 + 2 2}{2 ( 2 + 2 )}

Apply radical rule:

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

= \frac{2 3 + 2 2}{2 ^{\frac{1}{2}} ( 2 + 2 )}

Apply exponent rule:

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

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

= \frac{2 ^{- \frac{1}{2} + 1} 3 + 2 2}{2 + 2}

Subtract the numbers:

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

= \frac{2 ^{\frac{1}{2}} 3 + 2 2}{2 + 2}

Apply radical rule:

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

= \frac{2 3 + 2 2}{2 + 2}

= \frac{2 3 + 2 2}{2 + 2}

Factor

2 + 2 : 2 (2 + 1)

2 + 2

2 = 22

= 22 + 2

Factor out common term

2

= 2 (2 + 1)

= \frac{2 3 + 2 2}{2 ( 2 + 1 )}

Cancel the common factor:

2

= \frac{3 + 2 2}{2 + 1}

3 + 22 = 2 + 1

3 + 22

= 2 + 22 + 1

= (2)^{2} + 22 + (1)^{2}

1 = 1

1

Apply rule

1 = 1

= 1

= (2)^{2} + 22 + 1^{2}

22 \cdot 1 = 22

22 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 22

= (2)^{2} + 22 \cdot 1 + 1^{2}

Apply Perfect Square Formula:

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

(2)^{2} + 22 \cdot 1 + 1^{2} = (2 + 1)^{2}

= (2 + 1)^{2}

Apply radical rule:

(2 + 1)^{2} = 2 + 1

= 2 + 1

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

Apply rule

\frac{a}{a} = 1

= 1

= 1

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\frac{15.24}{tan ( 4 1 ^{\circ} )}

arctan(csc(pi/2))

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arctan (\frac{- 2}{6})

Frequently Asked Questions (FAQ)

What is the value of (tan((5pi)/8)-tan((3pi)/8))/(1+tan((5pi)/8)tan((3pi)/8)) ?
The value of (tan((5pi)/8)-tan((3pi)/8))/(1+tan((5pi)/8)tan((3pi)/8)) is 1