Trigonometry Examples

Verify the Identity 1/(csc(theta)-1)-1/(csc(theta)+1)=2tan(theta)^2
Step 1
Start on the left side.
Step 2
Subtract 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
Simplify each term.
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Step 3.1.1
Apply the distributive property.
Step 3.1.2
Multiply by .
Step 3.2
Subtract from .
Step 3.3
Add and .
Step 3.4
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.
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
Combine and .
Step 8
Rewrite as .
Step 9
Because the two sides have been shown to be equivalent, the equation is an identity.
is an identity