Trigonometry Examples

Solve for x tan(2x)=(2tan(x))/(1-tan(x)^2)
Step 1
Multiply both sides by .
Step 2
Simplify.
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Step 2.1
Simplify the left side.
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Step 2.1.1
Simplify .
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Step 2.1.1.1
Apply the distributive property.
Step 2.1.1.2
Simplify the expression.
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Step 2.1.1.2.1
Multiply by .
Step 2.1.1.2.2
Rewrite using the commutative property of multiplication.
Step 2.2
Simplify the right side.
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Step 2.2.1
Cancel the common factor of .
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Step 2.2.1.1
Cancel the common factor.
Step 2.2.1.2
Rewrite the expression.
Step 3
Solve for .
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Step 3.1
Simplify the left side.
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Step 3.1.1
Simplify each term.
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Step 3.1.1.1
Rewrite in terms of sines and cosines.
Step 3.1.1.2
Rewrite in terms of sines and cosines.
Step 3.1.1.3
Rewrite in terms of sines and cosines.
Step 3.1.1.4
Apply the product rule to .
Step 3.1.1.5
Multiply by .
Step 3.1.1.6
Simplify the numerator.
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Step 3.1.1.6.1
Apply the sine double-angle identity.
Step 3.1.1.6.2
Combine exponents.
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Step 3.1.1.6.2.1
Raise to the power of .
Step 3.1.1.6.2.2
Use the power rule to combine exponents.
Step 3.1.1.6.2.3
Add and .
Step 3.1.1.7
Use the double-angle identity to transform to .
Step 3.1.1.8
Cancel the common factor of and .
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Step 3.1.1.8.1
Factor out of .
Step 3.1.1.8.2
Cancel the common factors.
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Step 3.1.1.8.2.1
Factor out of .
Step 3.1.1.8.2.2
Cancel the common factor.
Step 3.1.1.8.2.3
Rewrite the expression.
Step 3.1.1.9
Apply the cosine double-angle identity.
Step 3.2
Simplify the right side.
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Step 3.2.1
Simplify .
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Step 3.2.1.1
Rewrite in terms of sines and cosines.
Step 3.2.1.2
Combine and .
Step 3.3
Multiply both sides of the equation by .
Step 3.4
Apply the distributive property.
Step 3.5
Cancel the common factor of .
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Step 3.5.1
Cancel the common factor.
Step 3.5.2
Rewrite the expression.
Step 3.6
Rewrite using the commutative property of multiplication.
Step 3.7
Cancel the common factor of .
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Step 3.7.1
Factor out of .
Step 3.7.2
Factor out of .
Step 3.7.3
Cancel the common factor.
Step 3.7.4
Rewrite the expression.
Step 3.8
Combine and .
Step 3.9
Move to the left of .
Step 3.10
Multiply both sides of the equation by .
Step 3.11
Apply the distributive property.
Step 3.12
Rewrite using the commutative property of multiplication.
Step 3.13
Simplify each term.
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Step 3.13.1
Cancel the common factor of .
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Step 3.13.1.1
Factor out of .
Step 3.13.1.2
Cancel the common factor.
Step 3.13.1.3
Rewrite the expression.
Step 3.13.2
Multiply by .
Step 3.14
Cancel the common factor of .
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Step 3.14.1
Cancel the common factor.
Step 3.14.2
Rewrite the expression.
Step 3.15
Use the double-angle identity to transform to .
Step 3.16
Subtract from both sides of the equation.
Step 3.17
Simplify the left side.
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Step 3.17.1
Simplify .
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Step 3.17.1.1
Simplify terms.
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Step 3.17.1.1.1
Simplify each term.
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Step 3.17.1.1.1.1
Apply the sine double-angle identity.
Step 3.17.1.1.1.2
Rewrite using the commutative property of multiplication.
Step 3.17.1.1.1.3
Multiply .
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Step 3.17.1.1.1.3.1
Raise to the power of .
Step 3.17.1.1.1.3.2
Raise to the power of .
Step 3.17.1.1.1.3.3
Use the power rule to combine exponents.
Step 3.17.1.1.1.3.4
Add and .
Step 3.17.1.1.1.4
Apply the distributive property.
Step 3.17.1.1.1.5
Multiply by .
Step 3.17.1.1.1.6
Multiply by .
Step 3.17.1.1.1.7
Apply the distributive property.
Step 3.17.1.1.1.8
Multiply by by adding the exponents.
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Step 3.17.1.1.1.8.1
Move .
Step 3.17.1.1.1.8.2
Multiply by .
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Step 3.17.1.1.1.8.2.1
Raise to the power of .
Step 3.17.1.1.1.8.2.2
Use the power rule to combine exponents.
Step 3.17.1.1.1.8.3
Add and .
Step 3.17.1.1.2
Simplify with factoring out.
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Step 3.17.1.1.2.1
Factor out of .
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Step 3.17.1.1.2.1.1
Factor out of .
Step 3.17.1.1.2.1.2
Factor out of .
Step 3.17.1.1.2.1.3
Factor out of .
Step 3.17.1.1.2.1.4
Factor out of .
Step 3.17.1.1.2.1.5
Factor out of .
Step 3.17.1.1.2.1.6
Factor out of .
Step 3.17.1.1.2.1.7
Factor out of .
Step 3.17.1.1.2.2
Simplify the expression.
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Step 3.17.1.1.2.2.1
Move .
Step 3.17.1.1.2.2.2
Reorder and .
Step 3.17.1.1.2.3
Rewrite as .
Step 3.17.1.1.2.4
Factor out of .
Step 3.17.1.1.2.5
Factor out of .
Step 3.17.1.1.2.6
Rewrite as .
Step 3.17.1.2
Apply pythagorean identity.
Step 3.17.1.3
Simplify by adding terms.
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Step 3.17.1.3.1
Subtract from .
Step 3.17.1.3.2
Add and .
Step 3.17.1.4
Multiply .
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Step 3.17.1.4.1
Multiply by .
Step 3.17.1.4.2
Multiply by .
Step 3.18
Since , the equation will always be true for any value of .
All real numbers
All real numbers
Step 4
The result can be shown in multiple forms.
All real numbers
Interval Notation: