Trigonometry Examples

Solve for x (2cot(2x))/(cos(x)-sin(x))=csc(x)+sec(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
Simplify .
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Step 2.1.1.1
Cancel the common factor of .
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Step 2.1.1.1.1
Cancel the common factor.
Step 2.1.1.1.2
Rewrite the expression.
Step 2.1.1.2
Rewrite in terms of sines and cosines.
Step 2.1.1.3
Combine and .
Step 2.1.1.4
Separate fractions.
Step 2.1.1.5
Convert from to .
Step 2.1.1.6
Divide by .
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 each term.
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Step 2.2.1.1.1
Rewrite in terms of sines and cosines.
Step 2.2.1.1.2
Rewrite in terms of sines and cosines.
Step 2.2.1.2
Expand using the FOIL Method.
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Step 2.2.1.2.1
Apply the distributive property.
Step 2.2.1.2.2
Apply the distributive property.
Step 2.2.1.2.3
Apply the distributive property.
Step 2.2.1.3
Simplify and combine like terms.
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Step 2.2.1.3.1
Simplify each term.
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Step 2.2.1.3.1.1
Combine and .
Step 2.2.1.3.1.2
Rewrite using the commutative property of multiplication.
Step 2.2.1.3.1.3
Cancel the common factor of .
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Step 2.2.1.3.1.3.1
Move the leading negative in into the numerator.
Step 2.2.1.3.1.3.2
Cancel the common factor.
Step 2.2.1.3.1.3.3
Rewrite the expression.
Step 2.2.1.3.1.4
Cancel the common factor of .
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Step 2.2.1.3.1.4.1
Cancel the common factor.
Step 2.2.1.3.1.4.2
Rewrite the expression.
Step 2.2.1.3.1.5
Rewrite using the commutative property of multiplication.
Step 2.2.1.3.1.6
Combine and .
Step 2.2.1.3.2
Add and .
Step 2.2.1.3.3
Add and .
Step 2.2.1.4
Simplify each term.
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Step 2.2.1.4.1
Convert from to .
Step 2.2.1.4.2
Convert from to .
Step 3
Solve for .
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Step 3.1
Simplify the left side.
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Step 3.1.1
Simplify .
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Step 3.1.1.1
Rewrite in terms of sines and cosines.
Step 3.1.1.2
Combine and .
Step 3.2
Simplify the right side.
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Step 3.2.1
Simplify each term.
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Step 3.2.1.1
Rewrite in terms of sines and cosines.
Step 3.2.1.2
Rewrite in terms of sines and cosines.
Step 3.3
Multiply both sides of the equation by .
Step 3.4
Cancel the common factor of .
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Step 3.4.1
Cancel the common factor.
Step 3.4.2
Rewrite the expression.
Step 3.5
Apply the distributive property.
Step 3.6
Combine and .
Step 3.7
Rewrite using the commutative property of multiplication.
Step 3.8
Simplify each term.
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Step 3.8.1
Simplify the numerator.
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Step 3.8.1.1
Apply the sine double-angle identity.
Step 3.8.1.2
Combine exponents.
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Step 3.8.1.2.1
Raise to the power of .
Step 3.8.1.2.2
Raise to the power of .
Step 3.8.1.2.3
Use the power rule to combine exponents.
Step 3.8.1.2.4
Add and .
Step 3.8.2
Cancel the common factor of .
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Step 3.8.2.1
Cancel the common factor.
Step 3.8.2.2
Divide by .
Step 3.8.3
Combine and .
Step 3.8.4
Simplify the numerator.
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Step 3.8.4.1
Apply the sine double-angle identity.
Step 3.8.4.2
Combine exponents.
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Step 3.8.4.2.1
Raise to the power of .
Step 3.8.4.2.2
Raise to the power of .
Step 3.8.4.2.3
Use the power rule to combine exponents.
Step 3.8.4.2.4
Add and .
Step 3.8.5
Cancel the common factor of .
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Step 3.8.5.1
Cancel the common factor.
Step 3.8.5.2
Divide by .
Step 3.8.6
Multiply by .
Step 3.9
Move all the expressions to the left side of the equation.
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Step 3.9.1
Subtract from both sides of the equation.
Step 3.9.2
Add to both sides of the equation.
Step 3.10
Use the double-angle identity to transform to .
Step 3.11
Simplify the left side.
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Step 3.11.1
Simplify .
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Step 3.11.1.1
Simplify with factoring out.
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Step 3.11.1.1.1
Factor out of .
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Step 3.11.1.1.1.1
Factor out of .
Step 3.11.1.1.1.2
Factor out of .
Step 3.11.1.1.1.3
Factor out of .
Step 3.11.1.1.1.4
Factor out of .
Step 3.11.1.1.2
Move .
Step 3.11.1.2
Apply pythagorean identity.
Step 3.11.1.3
Simplify by adding terms.
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Step 3.11.1.3.1
Subtract from .
Step 3.11.1.3.2
Add and .
Step 3.11.1.3.3
Multiply by .
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: