Trigonometry Examples

Expand Using Sum/Difference Formulas tan(75)
Step 1
First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, can be split into .
Step 2
Use the sum formula for tangent to simplify the expression. The formula states that .
Step 3
Simplify the numerator.
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Step 3.1
The exact value of is .
Step 3.2
The exact value of is .
Step 3.3
Write as a fraction with a common denominator.
Step 3.4
Combine the numerators over the common denominator.
Step 4
Simplify the denominator.
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Step 4.1
The exact value of is .
Step 4.2
The exact value of is .
Step 4.3
Multiply by .
Step 4.4
Write as a fraction with a common denominator.
Step 4.5
Combine the numerators over the common denominator.
Step 5
Multiply the numerator by the reciprocal of the denominator.
Step 6
Cancel the common factor of .
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Step 6.1
Cancel the common factor.
Step 6.2
Rewrite the expression.
Step 7
Multiply by .
Step 8
Simplify terms.
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Step 8.1
Multiply by .
Step 8.2
Expand the denominator using the FOIL method.
Step 8.3
Simplify.
Step 8.4
Apply the distributive property.
Step 8.5
Combine and .
Step 8.6
Cancel the common factor of .
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Step 8.6.1
Factor out of .
Step 8.6.2
Cancel the common factor.
Step 8.6.3
Rewrite the expression.
Step 9
Simplify each term.
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Step 9.1
Apply the distributive property.
Step 9.2
Move to the left of .
Step 9.3
Combine using the product rule for radicals.
Step 9.4
Simplify each term.
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Step 9.4.1
Multiply by .
Step 9.4.2
Rewrite as .
Step 9.4.3
Pull terms out from under the radical, assuming positive real numbers.
Step 9.5
Cancel the common factor of and .
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Step 9.5.1
Factor out of .
Step 9.5.2
Factor out of .
Step 9.5.3
Factor out of .
Step 9.5.4
Cancel the common factors.
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Step 9.5.4.1
Factor out of .
Step 9.5.4.2
Cancel the common factor.
Step 9.5.4.3
Rewrite the expression.
Step 10
Simplify terms.
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Step 10.1
Combine the numerators over the common denominator.
Step 10.2
Add and .
Step 10.3
Add and .
Step 10.4
Cancel the common factor of and .
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Step 10.4.1
Factor out of .
Step 10.4.2
Factor out of .
Step 10.4.3
Factor out of .
Step 10.4.4
Cancel the common factors.
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Step 10.4.4.1
Factor out of .
Step 10.4.4.2
Cancel the common factor.
Step 10.4.4.3
Rewrite the expression.
Step 10.4.4.4
Divide by .
Step 11
The result can be shown in multiple forms.
Exact Form:
Decimal Form: