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Popular Trigonometry >

tan(x)-sqrt(1-2tan^2(x))=0

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Solution

tan(x)−1−2tan2(x)​=0

Solution

x=0.52359…+πn
+1
Degrees
x=30∘+180∘n
Solution steps
tan(x)−1−2tan2(x)​=0
Solve by substitution
tan(x)−1−2tan2(x)​=0
Let: tan(x)=uu−1−2u2​=0
u−1−2u2​=0:u=31​​
u−1−2u2​=0
Remove square roots
u−1−2u2​=0
Subtract u from both sidesu−1−2u2​−u=0−u
Simplify−1−2u2​=−u
Square both sides:1−2u2=u2
u−1−2u2​=0
(−1−2u2​)2=(−u)2
Expand (−1−2u2​)2:1−2u2
(−1−2u2​)2
Apply exponent rule: (−a)n=an,if n is even(−1−2u2​)2=(1−2u2​)2=(1−2u2​)2
Apply radical rule: a​=a21​=((1−2u2)21​)2
Apply exponent rule: (ab)c=abc=(1−2u2)21​⋅2
21​⋅2=1
21​⋅2
Multiply fractions: a⋅cb​=ca⋅b​=21⋅2​
Cancel the common factor: 2=1
=1−2u2
Expand (−u)2:u2
(−u)2
Apply exponent rule: (−a)n=an,if n is even(−u)2=u2=u2
1−2u2=u2
1−2u2=u2
1−2u2=u2
Solve 1−2u2=u2:u=31​​,u=−31​​
1−2u2=u2
Move 1to the right side
1−2u2=u2
Subtract 1 from both sides1−2u2−1=u2−1
Simplify−2u2=u2−1
−2u2=u2−1
Move u2to the left side
−2u2=u2−1
Subtract u2 from both sides−2u2−u2=u2−1−u2
Simplify−3u2=−1
−3u2=−1
Divide both sides by −3
−3u2=−1
Divide both sides by −3−3−3u2​=−3−1​
Simplifyu2=31​
u2=31​
For x2=f(a) the solutions are x=f(a)​,−f(a)​
u=31​​,u=−31​​
u=31​​,u=−31​​
Verify Solutions:u=31​​True,u=−31​​False
Check the solutions by plugging them into u−1−2u2​=0
Remove the ones that don't agree with the equation.
Plug in u=31​​:True
31​​−1−2(31​​)2​=0
31​​−1−2(31​​)2​=0
31​​−1−2(31​​)2​
1−2(31​​)2​=31​​
1−2(31​​)2​
2(31​​)2=32​
2(31​​)2
(31​​)2=31​
(31​​)2
Apply radical rule: a​=a21​=((31​)21​)2
Apply exponent rule: (ab)c=abc=(31​)21​⋅2
21​⋅2=1
21​⋅2
Multiply fractions: a⋅cb​=ca⋅b​=21⋅2​
Cancel the common factor: 2=1
=31​
=2⋅31​
Multiply fractions: a⋅cb​=ca⋅b​=31⋅2​
Multiply the numbers: 1⋅2=2=32​
=1−32​​
Join 1−32​:31​
1−32​
Convert element to fraction: 1=31⋅3​=31⋅3​−32​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=31⋅3−2​
1⋅3−2=1
1⋅3−2
Multiply the numbers: 1⋅3=3=3−2
Subtract the numbers: 3−2=1=1
=31​
=31​​
=31​​−31​​
Add similar elements: 31​​−31​​=0=0
0=0
True
Plug in u=−31​​:False
(−31​​)−1−2(−31​​)2​=0
(−31​​)−1−2(−31​​)2​=−231​​
(−31​​)−1−2(−31​​)2​
Remove parentheses: (−a)=−a=−31​​−1−2(−31​​)2​
1−2(−31​​)2​=31​​
1−2(−31​​)2​
2(−31​​)2=32​
2(−31​​)2
(−31​​)2=31​
(−31​​)2
Apply exponent rule: (−a)n=an,if n is even(−31​​)2=(31​​)2=(31​​)2
Apply radical rule: a​=a21​=((31​)21​)2
Apply exponent rule: (ab)c=abc=(31​)21​⋅2
21​⋅2=1
21​⋅2
Multiply fractions: a⋅cb​=ca⋅b​=21⋅2​
Cancel the common factor: 2=1
=31​
=2⋅31​
Multiply fractions: a⋅cb​=ca⋅b​=31⋅2​
Multiply the numbers: 1⋅2=2=32​
=1−32​​
Join 1−32​:31​
1−32​
Convert element to fraction: 1=31⋅3​=31⋅3​−32​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=31⋅3−2​
1⋅3−2=1
1⋅3−2
Multiply the numbers: 1⋅3=3=3−2
Subtract the numbers: 3−2=1=1
=31​
=31​​
=−31​​−31​​
Add similar elements: −31​​−31​​=−231​​=−231​​
−231​​=0
False
The solution isu=31​​
Substitute back u=tan(x)tan(x)=31​​
tan(x)=31​​
tan(x)=31​​:x=arctan(31​​)+πn
tan(x)=31​​
Apply trig inverse properties
tan(x)=31​​
General solutions for tan(x)=31​​tan(x)=a⇒x=arctan(a)+πnx=arctan(31​​)+πn
x=arctan(31​​)+πn
Combine all the solutionsx=arctan(31​​)+πn
Show solutions in decimal formx=0.52359…+πn

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Frequently Asked Questions (FAQ)

  • What is the general solution for tan(x)-sqrt(1-2tan^2(x))=0 ?

    The general solution for tan(x)-sqrt(1-2tan^2(x))=0 is x=0.52359…+pin
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