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Popular Trigonometry >

(tan(45)-tan(30))/(1+tan(45)tan(30))

  • Pre Algebra
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Solution

1+tan(45∘)tan(30∘)tan(45∘)−tan(30∘)​

Solution

2−3​
+1
Decimal
0.26794…
Solution steps
1+tan(45∘)tan(30∘)tan(45∘)−tan(30∘)​
Use the following trivial identity:tan(45∘)=1
tan(45∘)
tan(x) periodicity table with 180∘n cycle:
x030∘45∘60∘90∘120∘135∘150∘​tan(x)033​​13​±∞−3​−1−33​​​​
=1
Use the following trivial identity:tan(30∘)=33​​
tan(30∘)
tan(x) periodicity table with 180∘n cycle:
x030∘45∘60∘90∘120∘135∘150∘​tan(x)033​​13​±∞−3​−1−33​​​​
=33​​
=1+1⋅33​​1−33​​​
Simplify 1+1⋅33​​1−33​​​:2−3​
1+1⋅33​​1−33​​​
Multiply: 1⋅33​​=33​​=1+33​​1−33​​​
Join 1+33​​:3​3​+1​
1+33​​
Convert element to fraction: 1=31⋅3​=31⋅3​+33​​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=31⋅3+3​​
Multiply the numbers: 1⋅3=3=33+3​​
Factor 3+3​:3​(3​+1)
3+3​
3=3​3​=3​3​+3​
Factor out common term 3​=3​(3​+1)
=33​(3​+1)​
Cancel 33​(3​+1)​:3​3​+1​
33​(3​+1)​
Apply radical rule: na​=an1​3​=321​=3321​(1+3​)​
Apply exponent rule: xbxa​=xb−a1​31321​​=31−21​1​=31−21​3​+1​
Subtract the numbers: 1−21​=21​=321​3​+1​
Apply radical rule: an1​=na​321​=3​=3​3​+1​
=3​3​+1​
=3​3​+1​1−33​​​
Join 1−33​​:3​3​−1​
1−33​​
Convert element to fraction: 1=31⋅3​=31⋅3​−33​​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=31⋅3−3​​
Multiply the numbers: 1⋅3=3=33−3​​
Factor 3−3​:3​(3​−1)
3−3​
3=3​3​=3​3​−3​
Factor out common term 3​=3​(3​−1)
=33​(3​−1)​
Cancel 33​(3​−1)​:3​3​−1​
33​(3​−1)​
Apply radical rule: na​=an1​3​=321​=3321​(3​−1)​
Apply exponent rule: xbxa​=xb−a1​31321​​=31−21​1​=31−21​3​−1​
Subtract the numbers: 1−21​=21​=321​3​−1​
Apply radical rule: an1​=na​321​=3​=3​3​−1​
=3​3​−1​
=3​3​+1​3​3​−1​​
Divide fractions: dc​ba​​=b⋅ca⋅d​=3​(3​+1)(3​−1)3​​
Cancel the common factor: 3​=3​+13​−1​
Rationalize 3​+13​−1​:2−3​
3​+13​−1​
Multiply by the conjugate 3​−13​−1​=(3​+1)(3​−1)(3​−1)(3​−1)​
(3​−1)(3​−1)=4−23​
(3​−1)(3​−1)
Apply exponent rule: ab⋅ac=ab+c(3​−1)(3​−1)=(3​−1)1+1=(3​−1)1+1
Add the numbers: 1+1=2=(3​−1)2
Apply Perfect Square Formula: (a−b)2=a2−2ab+b2a=3​,b=1
=(3​)2−23​⋅1+12
Simplify (3​)2−23​⋅1+12:4−23​
(3​)2−23​⋅1+12
Apply rule 1a=112=1=(3​)2−2⋅1⋅3​+1
(3​)2=3
(3​)2
Apply radical rule: a​=a21​=(321​)2
Apply exponent rule: (ab)c=abc=321​⋅2
21​⋅2=1
21​⋅2
Multiply fractions: a⋅cb​=ca⋅b​=21⋅2​
Cancel the common factor: 2=1
=3
23​⋅1=23​
23​⋅1
Multiply the numbers: 2⋅1=2=23​
=3−23​+1
Add the numbers: 3+1=4=4−23​
=4−23​
(3​+1)(3​−1)=2
(3​+1)(3​−1)
Apply Difference of Two Squares Formula: (a+b)(a−b)=a2−b2a=3​,b=1=(3​)2−12
Simplify (3​)2−12:2
(3​)2−12
Apply rule 1a=112=1=(3​)2−1
(3​)2=3
(3​)2
Apply radical rule: a​=a21​=(321​)2
Apply exponent rule: (ab)c=abc=321​⋅2
21​⋅2=1
21​⋅2
Multiply fractions: a⋅cb​=ca⋅b​=21⋅2​
Cancel the common factor: 2=1
=3
=3−1
Subtract the numbers: 3−1=2=2
=2
=24−23​​
Factor 4−23​:2(2−3​)
4−23​
Rewrite as=2⋅2−23​
Factor out common term 2=2(2−3​)
=22(2−3​)​
Divide the numbers: 22​=1=2−3​
=2−3​
=2−3​

Popular Examples

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Frequently Asked Questions (FAQ)

  • What is the value of (tan(45)-tan(30))/(1+tan(45)tan(30)) ?

    The value of (tan(45)-tan(30))/(1+tan(45)tan(30)) is 2-sqrt(3)
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