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Trigonometry Examples
Step 1
Step 1.1
Rewrite in terms of sines and cosines.
Step 2
Step 2.1
Simplify .
Step 2.1.1
Simplify the numerator.
Step 2.1.1.1
Rewrite as .
Step 2.1.1.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 2.1.1.3
Simplify.
Step 2.1.1.3.1
Rewrite in terms of sines and cosines.
Step 2.1.1.3.2
Rewrite in terms of sines and cosines.
Step 2.1.2
Rewrite in terms of sines and cosines.
Step 2.1.3
Combine and .
Step 2.1.4
Multiply the numerator by the reciprocal of the denominator.
Step 2.1.5
Expand using the FOIL Method.
Step 2.1.5.1
Apply the distributive property.
Step 2.1.5.2
Apply the distributive property.
Step 2.1.5.3
Apply the distributive property.
Step 2.1.6
Simplify and combine like terms.
Step 2.1.6.1
Simplify each term.
Step 2.1.6.1.1
Multiply .
Step 2.1.6.1.1.1
Multiply by .
Step 2.1.6.1.1.2
Raise to the power of .
Step 2.1.6.1.1.3
Raise to the power of .
Step 2.1.6.1.1.4
Use the power rule to combine exponents.
Step 2.1.6.1.1.5
Add and .
Step 2.1.6.1.1.6
Raise to the power of .
Step 2.1.6.1.1.7
Raise to the power of .
Step 2.1.6.1.1.8
Use the power rule to combine exponents.
Step 2.1.6.1.1.9
Add and .
Step 2.1.6.1.2
Combine and .
Step 2.1.6.1.3
Move to the left of .
Step 2.1.6.1.4
Move the negative in front of the fraction.
Step 2.1.6.1.5
Multiply by .
Step 2.1.6.1.6
Multiply by .
Step 2.1.6.2
Add and .
Step 2.1.6.3
Add and .
Step 2.1.7
Simplify terms.
Step 2.1.7.1
Apply the distributive property.
Step 2.1.7.2
Combine.
Step 2.1.7.3
Rewrite as .
Step 2.1.8
Simplify each term.
Step 2.1.8.1
Cancel the common factor of and .
Step 2.1.8.1.1
Factor out of .
Step 2.1.8.1.2
Cancel the common factors.
Step 2.1.8.1.2.1
Factor out of .
Step 2.1.8.1.2.2
Cancel the common factor.
Step 2.1.8.1.2.3
Rewrite the expression.
Step 2.1.8.2
Cancel the common factor of and .
Step 2.1.8.2.1
Factor out of .
Step 2.1.8.2.2
Cancel the common factors.
Step 2.1.8.2.2.1
Factor out of .
Step 2.1.8.2.2.2
Cancel the common factor.
Step 2.1.8.2.2.3
Rewrite the expression.
Step 2.1.8.3
Move to the left of .
Step 3
Multiply both sides of the equation by .
Step 4
Step 4.1
Cancel the common factor.
Step 4.2
Rewrite the expression.
Step 5
Apply the distributive property.
Step 6
Combine and .
Step 7
Rewrite using the commutative property of multiplication.
Step 8
Step 8.1
Simplify the numerator.
Step 8.1.1
Apply the sine double-angle identity.
Step 8.1.2
Combine exponents.
Step 8.1.2.1
Raise to the power of .
Step 8.1.2.2
Raise to the power of .
Step 8.1.2.3
Use the power rule to combine exponents.
Step 8.1.2.4
Add and .
Step 8.2
Cancel the common factor of .
Step 8.2.1
Cancel the common factor.
Step 8.2.2
Rewrite the expression.
Step 8.3
Cancel the common factor of .
Step 8.3.1
Cancel the common factor.
Step 8.3.2
Divide by .
Step 8.4
Combine and .
Step 8.5
Simplify the numerator.
Step 8.5.1
Apply the sine double-angle identity.
Step 8.5.2
Combine exponents.
Step 8.5.2.1
Raise to the power of .
Step 8.5.2.2
Raise to the power of .
Step 8.5.2.3
Use the power rule to combine exponents.
Step 8.5.2.4
Add and .
Step 8.6
Cancel the common factor of .
Step 8.6.1
Cancel the common factor.
Step 8.6.2
Rewrite the expression.
Step 8.7
Cancel the common factor of .
Step 8.7.1
Cancel the common factor.
Step 8.7.2
Divide by .
Step 9
Apply the cosine double-angle identity.
Step 10
Step 10.1
Subtract from both sides of the equation.
Step 10.2
Subtract from .
Step 11
Since , the equation will always be true for any value of .
All real numbers
Step 12
The result can be shown in multiple forms.
All real numbers
Interval Notation: