Trigonometry Examples

Solve for x (tan(x)-1)/(tan(x)+1)=(1-cot(x))/(1+cot(x))
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
Cancel the common factor of .
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Step 2.1.1.1
Cancel the common factor.
Step 2.1.1.2
Rewrite the expression.
Step 2.2
Simplify the right side.
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Step 2.2.1
Simplify .
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Step 2.2.1.1
Simplify terms.
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Step 2.2.1.1.1
Apply the distributive property.
Step 2.2.1.1.2
Combine and .
Step 2.2.1.1.3
Multiply by .
Step 2.2.1.1.4
Combine the numerators over the common denominator.
Step 2.2.1.2
Simplify each term.
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Step 2.2.1.2.1
Apply the distributive property.
Step 2.2.1.2.2
Multiply by .
Step 2.2.1.2.3
Rewrite in terms of sines and cosines, then cancel the common factors.
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Step 2.2.1.2.3.1
Add parentheses.
Step 2.2.1.2.3.2
Rewrite in terms of sines and cosines.
Step 2.2.1.2.3.3
Cancel the common factors.
Step 2.2.1.2.4
Multiply by .
Step 2.2.1.3
Combine the opposite terms in .
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Step 2.2.1.3.1
Add and .
Step 2.2.1.3.2
Add and .
Step 3
Solve for .
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Step 3.1
Simplify the right side.
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Step 3.1.1
Combine the numerators over the common denominator.
Step 3.2
Simplify the left side.
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Step 3.2.1
Rewrite in terms of sines and cosines.
Step 3.3
Simplify the right side.
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Step 3.3.1
Simplify each term.
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Step 3.3.1.1
Simplify the numerator.
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Step 3.3.1.1.1
Rewrite in terms of sines and cosines.
Step 3.3.1.1.2
Rewrite in terms of sines and cosines.
Step 3.3.1.2
Rewrite in terms of sines and cosines.
Step 3.3.1.3
Multiply the numerator and denominator of the fraction by .
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Step 3.3.1.3.1
Multiply by .
Step 3.3.1.3.2
Combine.
Step 3.3.1.4
Apply the distributive property.
Step 3.3.1.5
Simplify by cancelling.
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Step 3.3.1.5.1
Cancel the common factor of .
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Step 3.3.1.5.1.1
Factor out of .
Step 3.3.1.5.1.2
Cancel the common factor.
Step 3.3.1.5.1.3
Rewrite the expression.
Step 3.3.1.5.2
Raise to the power of .
Step 3.3.1.5.3
Raise to the power of .
Step 3.3.1.5.4
Use the power rule to combine exponents.
Step 3.3.1.5.5
Add and .
Step 3.3.1.5.6
Cancel the common factor of .
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Step 3.3.1.5.6.1
Move the leading negative in into the numerator.
Step 3.3.1.5.6.2
Factor out of .
Step 3.3.1.5.6.3
Cancel the common factor.
Step 3.3.1.5.6.4
Rewrite the expression.
Step 3.3.1.5.7
Raise to the power of .
Step 3.3.1.5.8
Raise to the power of .
Step 3.3.1.5.9
Use the power rule to combine exponents.
Step 3.3.1.5.10
Add and .
Step 3.3.1.5.11
Cancel the common factor of .
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Step 3.3.1.5.11.1
Factor out of .
Step 3.3.1.5.11.2
Cancel the common factor.
Step 3.3.1.5.11.3
Rewrite the expression.
Step 3.3.1.5.12
Raise to the power of .
Step 3.3.1.5.13
Raise to the power of .
Step 3.3.1.5.14
Use the power rule to combine exponents.
Step 3.3.1.5.15
Add and .
Step 3.3.1.6
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 3.3.1.7
Simplify the denominator.
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Step 3.3.1.7.1
Factor out of .
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Step 3.3.1.7.1.1
Factor out of .
Step 3.3.1.7.1.2
Factor out of .
Step 3.3.1.7.1.3
Factor out of .
Step 3.3.1.7.2
Multiply by .
Step 3.3.1.8
Cancel the common factor of .
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Step 3.3.1.8.1
Cancel the common factor.
Step 3.3.1.8.2
Rewrite the expression.
Step 3.4
Multiply both sides of the equation by .
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
Apply the distributive property.
Step 3.7
Cancel the common factor of .
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Step 3.7.1
Cancel the common factor.
Step 3.7.2
Rewrite the expression.
Step 3.8
Multiply by .
Step 3.9
Combine the opposite terms in .
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Step 3.9.1
Add and .
Step 3.9.2
Add and .
Step 3.10
For the two functions to be equal, the arguments of each must be equal.
Step 3.11
Move all terms containing to the left side of the equation.
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Step 3.11.1
Subtract from both sides of the equation.
Step 3.11.2
Subtract from .
Step 3.12
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: