Trigonometry Examples

Simplify (sec(x)^2-cos(x)^2)/(tan(x)^2)
Step 1
Simplify the numerator.
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Step 1.1
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 1.2
Simplify.
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Step 1.2.1
Rewrite in terms of sines and cosines.
Step 1.2.2
Rewrite in terms of sines and cosines.
Step 2
Simplify the denominator.
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Step 2.1
Rewrite in terms of sines and cosines.
Step 2.2
Apply the product rule to .
Step 3
Multiply the numerator by the reciprocal of the denominator.
Step 4
Expand using the FOIL Method.
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Step 4.1
Apply the distributive property.
Step 4.2
Apply the distributive property.
Step 4.3
Apply the distributive property.
Step 5
Simplify terms.
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Step 5.1
Combine the opposite terms in .
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Step 5.1.1
Reorder the factors in the terms and .
Step 5.1.2
Add and .
Step 5.1.3
Add and .
Step 5.2
Simplify each term.
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Step 5.2.1
Multiply .
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Step 5.2.1.1
Multiply by .
Step 5.2.1.2
Raise to the power of .
Step 5.2.1.3
Raise to the power of .
Step 5.2.1.4
Use the power rule to combine exponents.
Step 5.2.1.5
Add and .
Step 5.2.2
Rewrite using the commutative property of multiplication.
Step 5.2.3
Multiply .
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Step 5.2.3.1
Raise to the power of .
Step 5.2.3.2
Raise to the power of .
Step 5.2.3.3
Use the power rule to combine exponents.
Step 5.2.3.4
Add and .
Step 5.3
Simplify terms.
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Step 5.3.1
Apply the distributive property.
Step 5.3.2
Combine.
Step 6
Multiply .
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Step 6.1
Combine and .
Step 6.2
Multiply by by adding the exponents.
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Step 6.2.1
Use the power rule to combine exponents.
Step 6.2.2
Add and .
Step 7
Cancel the common factor of .
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Step 7.1
Cancel the common factor.
Step 7.2
Rewrite the expression.
Step 8
Rewrite as .
Step 9
Rewrite as .
Step 10
Rewrite as .
Step 11
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 12
Simplify terms.
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Step 12.1
Simplify each term.
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Step 12.1.1
Convert from to .
Step 12.1.2
Factor out of .
Step 12.1.3
Separate fractions.
Step 12.1.4
Convert from to .
Step 12.1.5
Divide by .
Step 12.2
Combine the numerators over the common denominator.
Step 13
Apply pythagorean identity.
Step 14
Simplify terms.
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Step 14.1
Cancel the common factor of and .
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Step 14.1.1
Factor out of .
Step 14.1.2
Cancel the common factors.
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Step 14.1.2.1
Multiply by .
Step 14.1.2.2
Cancel the common factor.
Step 14.1.2.3
Rewrite the expression.
Step 14.1.2.4
Divide by .
Step 14.2
Apply the distributive property.
Step 15
Rewrite in terms of sines and cosines, then cancel the common factors.
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Step 15.1
Rewrite in terms of sines and cosines.
Step 15.2
Cancel the common factors.