Trigonometry Examples

Verify the Identity (1+cos(x))/(sin(x))+(sin(x))/(1+cos(x))=2csc(x)
Step 1
Start on the left side.
Step 2
Simplify the expression.
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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 2.5
Simplify the numerator.
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Step 2.5.1
Expand using the FOIL Method.
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Step 2.5.1.1
Apply the distributive property.
Step 2.5.1.2
Apply the distributive property.
Step 2.5.1.3
Apply the distributive property.
Step 2.5.2
Simplify and combine like terms.
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Step 2.5.2.1
Simplify each term.
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Step 2.5.2.1.1
Multiply by .
Step 2.5.2.1.2
Multiply by .
Step 2.5.2.1.3
Multiply by .
Step 2.5.2.1.4
Multiply .
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Step 2.5.2.1.4.1
Raise to the power of .
Step 2.5.2.1.4.2
Raise to the power of .
Step 2.5.2.1.4.3
Use the power rule to combine exponents.
Step 2.5.2.1.4.4
Add and .
Step 2.5.2.2
Add and .
Step 2.5.3
Multiply .
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Step 2.5.3.1
Raise to the power of .
Step 2.5.3.2
Raise to the power of .
Step 2.5.3.3
Use the power rule to combine exponents.
Step 2.5.3.4
Add and .
Step 2.5.4
Rewrite in a factored form.
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Step 2.5.4.1
Rearrange terms.
Step 2.5.4.2
Apply pythagorean identity.
Step 2.5.4.3
Add and .
Step 2.5.4.4
Factor out of .
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Step 2.5.4.4.1
Factor out of .
Step 2.5.4.4.2
Factor out of .
Step 2.6
Cancel the common factor of .
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Step 2.6.1
Cancel the common factor.
Step 2.6.2
Rewrite the expression.
Step 3
Rewrite as .
Step 4
Because the two sides have been shown to be equivalent, the equation is an identity.
is an identity