Trigonometry Examples

Verify the Identity 1/(1+sec(x))+1/(1-sec(x))=-2cot(x)^2
Step 1
Start on the left side.
Step 2
Add fractions.
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Step 2.1
To write as a fraction with a common denominator, multiply by .
Step 2.2
To write as a fraction with a common denominator, multiply by .
Step 2.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 2.3.1
Multiply by .
Step 2.3.2
Multiply by .
Step 2.3.3
Reorder the factors of .
Step 2.4
Combine the numerators over the common denominator.
Step 3
Simplify numerator.
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Step 3.1
Add and .
Step 3.2
Add and .
Step 3.3
Add and .
Step 4
Simplify denominator.
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Step 4.1
Expand using the FOIL Method.
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Step 4.1.1
Apply the distributive property.
Step 4.1.2
Apply the distributive property.
Step 4.1.3
Apply the distributive property.
Step 4.2
Simplify and combine like terms.
Step 5
Apply Pythagorean identity.
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Step 5.1
Reorder and .
Step 5.2
Factor out of .
Step 5.3
Rewrite as .
Step 5.4
Factor out of .
Step 5.5
Apply pythagorean identity.
Step 6
Convert to sines and cosines.
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Step 6.1
Write in sines and cosines using the quotient identity.
Step 6.2
Apply the product rule to .
Step 7
Simplify.
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Step 7.1
Multiply the numerator by the reciprocal of the denominator.
Step 7.2
Multiply .
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Step 7.2.1
Multiply by .
Step 7.2.2
Combine and .
Step 7.3
Move the negative in front of the fraction.
Step 8
Write as a fraction with denominator .
Step 9
Combine.
Step 10
Multiply by .
Step 11
Multiply by .
Step 12
Move the negative in front of the fraction.
Step 13
Now consider the right side of the equation.
Step 14
Convert to sines and cosines.
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Step 14.1
Write in sines and cosines using the quotient identity.
Step 14.2
Apply the product rule to .
Step 15
Simplify.
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Step 15.1
Combine and .
Step 15.2
Move the negative in front of the fraction.
Step 16
Because the two sides have been shown to be equivalent, the equation is an identity.
is an identity