Trigonometry Examples

Find the Inverse cos(arccsc(u))
Step 1
Interchange the variables.
Step 2
Solve for .
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Step 2.1
Rewrite the equation as .
Step 2.2
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 2.3
Take the inverse arccosecant of both sides of the equation to extract from inside the arccosecant.
Step 2.4
Simplify the right side.
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Step 2.4.1
Simplify .
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Step 2.4.1.1
Draw a triangle in the plane with vertices , , and the origin. Then is the angle between the positive x-axis and the ray beginning at the origin and passing through . Therefore, is .
Step 2.4.1.2
Simplify the denominator.
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Step 2.4.1.2.1
Rewrite as .
Step 2.4.1.2.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 2.4.1.3
Multiply by .
Step 2.4.1.4
Combine and simplify the denominator.
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Step 2.4.1.4.1
Multiply by .
Step 2.4.1.4.2
Raise to the power of .
Step 2.4.1.4.3
Raise to the power of .
Step 2.4.1.4.4
Use the power rule to combine exponents.
Step 2.4.1.4.5
Add and .
Step 2.4.1.4.6
Rewrite as .
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Step 2.4.1.4.6.1
Use to rewrite as .
Step 2.4.1.4.6.2
Apply the power rule and multiply exponents, .
Step 2.4.1.4.6.3
Combine and .
Step 2.4.1.4.6.4
Cancel the common factor of .
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Step 2.4.1.4.6.4.1
Cancel the common factor.
Step 2.4.1.4.6.4.2
Rewrite the expression.
Step 2.4.1.4.6.5
Simplify.
Step 3
Replace with to show the final answer.
Step 4
Verify if is the inverse of .
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Step 4.1
To verify the inverse, check if and .
Step 4.2
Evaluate .
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Step 4.2.1
Set up the composite result function.
Step 4.2.2
Evaluate by substituting in the value of into .
Step 4.2.3
Remove parentheses.
Step 4.2.4
Simplify the numerator.
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Step 4.2.4.1
Draw a triangle in the plane with vertices , , and the origin. Then is the angle between the positive x-axis and the ray beginning at the origin and passing through . Therefore, is .
Step 4.2.4.2
Simplify the numerator.
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Step 4.2.4.2.1
Rewrite as .
Step 4.2.4.2.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.2.4.3
Write as a fraction with a common denominator.
Step 4.2.4.4
Combine the numerators over the common denominator.
Step 4.2.4.5
Draw a triangle in the plane with vertices , , and the origin. Then is the angle between the positive x-axis and the ray beginning at the origin and passing through . Therefore, is .
Step 4.2.4.6
Simplify the numerator.
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Step 4.2.4.6.1
Rewrite as .
Step 4.2.4.6.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.2.4.7
Write as a fraction with a common denominator.
Step 4.2.4.8
Combine the numerators over the common denominator.
Step 4.2.4.9
Multiply by .
Step 4.2.4.10
Multiply by .
Step 4.2.4.11
Expand using the FOIL Method.
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Step 4.2.4.11.1
Apply the distributive property.
Step 4.2.4.11.2
Apply the distributive property.
Step 4.2.4.11.3
Apply the distributive property.
Step 4.2.4.12
Combine the opposite terms in .
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Step 4.2.4.12.1
Reorder the factors in the terms and .
Step 4.2.4.12.2
Add and .
Step 4.2.4.12.3
Add and .
Step 4.2.4.13
Simplify each term.
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Step 4.2.4.13.1
Multiply by .
Step 4.2.4.13.2
Rewrite using the commutative property of multiplication.
Step 4.2.4.13.3
Multiply .
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Step 4.2.4.13.3.1
Raise to the power of .
Step 4.2.4.13.3.2
Raise to the power of .
Step 4.2.4.13.3.3
Use the power rule to combine exponents.
Step 4.2.4.13.3.4
Add and .
Step 4.2.4.13.4
Rewrite as .
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Step 4.2.4.13.4.1
Use to rewrite as .
Step 4.2.4.13.4.2
Apply the power rule and multiply exponents, .
Step 4.2.4.13.4.3
Combine and .
Step 4.2.4.13.4.4
Cancel the common factor of .
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Step 4.2.4.13.4.4.1
Cancel the common factor.
Step 4.2.4.13.4.4.2
Rewrite the expression.
Step 4.2.4.13.4.5
Simplify.
Step 4.2.4.13.5
Expand using the FOIL Method.
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Step 4.2.4.13.5.1
Apply the distributive property.
Step 4.2.4.13.5.2
Apply the distributive property.
Step 4.2.4.13.5.3
Apply the distributive property.
Step 4.2.4.13.6
Simplify and combine like terms.
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Step 4.2.4.13.6.1
Simplify each term.
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Step 4.2.4.13.6.1.1
Multiply by .
Step 4.2.4.13.6.1.2
Move to the left of .
Step 4.2.4.13.6.1.3
Rewrite as .
Step 4.2.4.13.6.1.4
Multiply by .
Step 4.2.4.13.6.1.5
Multiply by .
Step 4.2.4.13.6.2
Add and .
Step 4.2.4.13.6.3
Add and .
Step 4.2.4.13.7
Apply the distributive property.
Step 4.2.4.13.8
Multiply by .
Step 4.2.4.14
Subtract from .
Step 4.2.4.15
Add and .
Step 4.2.4.16
Rewrite as .
Step 4.2.4.17
Rewrite as .
Step 4.2.4.18
Pull terms out from under the radical, assuming positive real numbers.
Step 4.2.5
Simplify the denominator.
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Step 4.2.5.1
Draw a triangle in the plane with vertices , , and the origin. Then is the angle between the positive x-axis and the ray beginning at the origin and passing through . Therefore, is .
Step 4.2.5.2
Simplify the numerator.
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Step 4.2.5.2.1
Rewrite as .
Step 4.2.5.2.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.2.5.3
Write as a fraction with a common denominator.
Step 4.2.5.4
Combine the numerators over the common denominator.
Step 4.2.5.5
Draw a triangle in the plane with vertices , , and the origin. Then is the angle between the positive x-axis and the ray beginning at the origin and passing through . Therefore, is .
Step 4.2.5.6
Simplify the numerator.
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Step 4.2.5.6.1
Rewrite as .
Step 4.2.5.6.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.2.5.7
Write as a fraction with a common denominator.
Step 4.2.5.8
Combine the numerators over the common denominator.
Step 4.2.6
Multiply by .
Step 4.2.7
Simplify the denominator.
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Step 4.2.7.1
Use the power rule to combine exponents.
Step 4.2.7.2
Add and .
Step 4.2.8
Simplify the denominator.
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Step 4.2.8.1
Expand using the FOIL Method.
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Step 4.2.8.1.1
Apply the distributive property.
Step 4.2.8.1.2
Apply the distributive property.
Step 4.2.8.1.3
Apply the distributive property.
Step 4.2.8.2
Combine the opposite terms in .
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Step 4.2.8.2.1
Reorder the factors in the terms and .
Step 4.2.8.2.2
Add and .
Step 4.2.8.2.3
Add and .
Step 4.2.8.3
Simplify each term.
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Step 4.2.8.3.1
Multiply by .
Step 4.2.8.3.2
Rewrite using the commutative property of multiplication.
Step 4.2.8.3.3
Multiply .
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Step 4.2.8.3.3.1
Raise to the power of .
Step 4.2.8.3.3.2
Raise to the power of .
Step 4.2.8.3.3.3
Use the power rule to combine exponents.
Step 4.2.8.3.3.4
Add and .
Step 4.2.8.3.4
Rewrite as .
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Step 4.2.8.3.4.1
Use to rewrite as .
Step 4.2.8.3.4.2
Apply the power rule and multiply exponents, .
Step 4.2.8.3.4.3
Combine and .
Step 4.2.8.3.4.4
Cancel the common factor of .
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Step 4.2.8.3.4.4.1
Cancel the common factor.
Step 4.2.8.3.4.4.2
Rewrite the expression.
Step 4.2.8.3.4.5
Simplify.
Step 4.2.8.3.5
Expand using the FOIL Method.
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Step 4.2.8.3.5.1
Apply the distributive property.
Step 4.2.8.3.5.2
Apply the distributive property.
Step 4.2.8.3.5.3
Apply the distributive property.
Step 4.2.8.3.6
Simplify and combine like terms.
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Step 4.2.8.3.6.1
Simplify each term.
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Step 4.2.8.3.6.1.1
Multiply by .
Step 4.2.8.3.6.1.2
Move to the left of .
Step 4.2.8.3.6.1.3
Rewrite as .
Step 4.2.8.3.6.1.4
Multiply by .
Step 4.2.8.3.6.1.5
Multiply by .
Step 4.2.8.3.6.2
Add and .
Step 4.2.8.3.6.3
Add and .
Step 4.2.8.3.7
Apply the distributive property.
Step 4.2.8.3.8
Multiply by .
Step 4.2.8.4
Subtract from .
Step 4.2.8.5
Add and .
Step 4.2.9
Multiply the numerator by the reciprocal of the denominator.
Step 4.2.10
Cancel the common factor of .
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Step 4.2.10.1
Factor out of .
Step 4.2.10.2
Cancel the common factor.
Step 4.2.10.3
Rewrite the expression.
Step 4.3
Evaluate .
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Step 4.3.1
Set up the composite result function.
Step 4.3.2
Evaluate by substituting in the value of into .
Step 4.3.3
Draw a triangle in the plane with vertices , , and the origin. Then is the angle between the positive x-axis and the ray beginning at the origin and passing through . Therefore, is .
Step 4.3.4
Multiply the numerator by the reciprocal of the denominator.
Step 4.3.5
Rewrite as .
Step 4.3.6
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.3.7
Simplify.
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Step 4.3.7.1
Write as a fraction with a common denominator.
Step 4.3.7.2
Combine the numerators over the common denominator.
Step 4.3.7.3
Rewrite in a factored form.
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Step 4.3.7.3.1
Use to rewrite as .
Step 4.3.7.3.2
Rewrite as .
Step 4.3.7.3.3
Let . Substitute for all occurrences of .
Step 4.3.7.3.4
Factor out of .
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Step 4.3.7.3.4.1
Raise to the power of .
Step 4.3.7.3.4.2
Factor out of .
Step 4.3.7.3.4.3
Factor out of .
Step 4.3.7.3.4.4
Factor out of .
Step 4.3.7.3.5
Replace all occurrences of with .
Step 4.3.7.3.6
Simplify.
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Step 4.3.7.3.6.1
Expand using the FOIL Method.
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Step 4.3.7.3.6.1.1
Apply the distributive property.
Step 4.3.7.3.6.1.2
Apply the distributive property.
Step 4.3.7.3.6.1.3
Apply the distributive property.
Step 4.3.7.3.6.2
Simplify and combine like terms.
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Step 4.3.7.3.6.2.1
Simplify each term.
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Step 4.3.7.3.6.2.1.1
Multiply by .
Step 4.3.7.3.6.2.1.2
Multiply by .
Step 4.3.7.3.6.2.1.3
Multiply by .
Step 4.3.7.3.6.2.1.4
Rewrite using the commutative property of multiplication.
Step 4.3.7.3.6.2.1.5
Multiply by by adding the exponents.
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Step 4.3.7.3.6.2.1.5.1
Move .
Step 4.3.7.3.6.2.1.5.2
Multiply by .
Step 4.3.7.3.6.2.2
Add and .
Step 4.3.7.3.6.2.3
Add and .
Step 4.3.7.3.6.3
Simplify each term.
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Step 4.3.7.3.6.3.1
Expand using the FOIL Method.
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Step 4.3.7.3.6.3.1.1
Apply the distributive property.
Step 4.3.7.3.6.3.1.2
Apply the distributive property.
Step 4.3.7.3.6.3.1.3
Apply the distributive property.
Step 4.3.7.3.6.3.2
Simplify and combine like terms.
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Step 4.3.7.3.6.3.2.1
Simplify each term.
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Step 4.3.7.3.6.3.2.1.1
Multiply by .
Step 4.3.7.3.6.3.2.1.2
Multiply by .
Step 4.3.7.3.6.3.2.1.3
Multiply by .
Step 4.3.7.3.6.3.2.1.4
Rewrite using the commutative property of multiplication.
Step 4.3.7.3.6.3.2.1.5
Multiply by by adding the exponents.
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Step 4.3.7.3.6.3.2.1.5.1
Move .
Step 4.3.7.3.6.3.2.1.5.2
Multiply by .
Step 4.3.7.3.6.3.2.2
Add and .
Step 4.3.7.3.6.3.2.3
Add and .
Step 4.3.7.4
To write as a fraction with a common denominator, multiply by .
Step 4.3.7.5
Combine and .
Step 4.3.7.6
Combine the numerators over the common denominator.
Step 4.3.7.7
Rewrite in a factored form.
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Step 4.3.7.7.1
Use to rewrite as .
Step 4.3.7.7.2
Rewrite as .
Step 4.3.7.7.3
Let . Substitute for all occurrences of .
Step 4.3.7.7.4
Factor out of .
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Step 4.3.7.7.4.1
Raise to the power of .
Step 4.3.7.7.4.2
Factor out of .
Step 4.3.7.7.4.3
Factor out of .
Step 4.3.7.7.4.4
Factor out of .
Step 4.3.7.7.5
Replace all occurrences of with .
Step 4.3.7.7.6
Simplify.
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Step 4.3.7.7.6.1
Expand using the FOIL Method.
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Step 4.3.7.7.6.1.1
Apply the distributive property.
Step 4.3.7.7.6.1.2
Apply the distributive property.
Step 4.3.7.7.6.1.3
Apply the distributive property.
Step 4.3.7.7.6.2
Simplify and combine like terms.
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Step 4.3.7.7.6.2.1
Simplify each term.
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Step 4.3.7.7.6.2.1.1
Multiply by .
Step 4.3.7.7.6.2.1.2
Multiply by .
Step 4.3.7.7.6.2.1.3
Multiply by .
Step 4.3.7.7.6.2.1.4
Rewrite using the commutative property of multiplication.
Step 4.3.7.7.6.2.1.5
Multiply by by adding the exponents.
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Step 4.3.7.7.6.2.1.5.1
Move .
Step 4.3.7.7.6.2.1.5.2
Multiply by .
Step 4.3.7.7.6.2.2
Add and .
Step 4.3.7.7.6.2.3
Add and .
Step 4.3.7.7.6.3
Simplify each term.
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Step 4.3.7.7.6.3.1
Expand using the FOIL Method.
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Step 4.3.7.7.6.3.1.1
Apply the distributive property.
Step 4.3.7.7.6.3.1.2
Apply the distributive property.
Step 4.3.7.7.6.3.1.3
Apply the distributive property.
Step 4.3.7.7.6.3.2
Simplify and combine like terms.
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Step 4.3.7.7.6.3.2.1
Simplify each term.
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Step 4.3.7.7.6.3.2.1.1
Multiply by .
Step 4.3.7.7.6.3.2.1.2
Multiply by .
Step 4.3.7.7.6.3.2.1.3
Multiply by .
Step 4.3.7.7.6.3.2.1.4
Rewrite using the commutative property of multiplication.
Step 4.3.7.7.6.3.2.1.5
Multiply by by adding the exponents.
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Step 4.3.7.7.6.3.2.1.5.1
Move .
Step 4.3.7.7.6.3.2.1.5.2
Multiply by .
Step 4.3.7.7.6.3.2.2
Add and .
Step 4.3.7.7.6.3.2.3
Add and .
Step 4.3.8
Multiply by .
Step 4.3.9
Combine exponents.
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Step 4.3.9.1
Multiply by by adding the exponents.
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Step 4.3.9.1.1
Move .
Step 4.3.9.1.2
Use the power rule to combine exponents.
Step 4.3.9.1.3
Combine the numerators over the common denominator.
Step 4.3.9.1.4
Add and .
Step 4.3.9.1.5
Divide by .
Step 4.3.9.2
Simplify .
Step 4.3.10
Combine exponents.
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Step 4.3.10.1
Raise to the power of .
Step 4.3.10.2
Raise to the power of .
Step 4.3.10.3
Use the power rule to combine exponents.
Step 4.3.10.4
Add and .
Step 4.3.10.5
Raise to the power of .
Step 4.3.10.6
Raise to the power of .
Step 4.3.10.7
Use the power rule to combine exponents.
Step 4.3.10.8
Add and .
Step 4.3.11
Simplify the numerator.
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Step 4.3.11.1
Rewrite as .
Step 4.3.11.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.3.12
Factor out of .
Step 4.3.13
Cancel the common factors.
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Step 4.3.13.1
Factor out of .
Step 4.3.13.2
Cancel the common factor.
Step 4.3.13.3
Rewrite the expression.
Step 4.3.14
Factor out of .
Step 4.3.15
Cancel the common factors.
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Step 4.3.15.1
Factor out of .
Step 4.3.15.2
Cancel the common factor.
Step 4.3.15.3
Rewrite the expression.
Step 4.3.16
Rewrite as .
Step 4.3.17
Combine.
Step 4.3.18
Simplify the denominator.
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Step 4.3.18.1
Raise to the power of .
Step 4.3.18.2
Raise to the power of .
Step 4.3.18.3
Use the power rule to combine exponents.
Step 4.3.18.4
Add and .
Step 4.3.19
Simplify the denominator.
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Step 4.3.19.1
Rewrite as .
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Step 4.3.19.1.1
Use to rewrite as .
Step 4.3.19.1.2
Apply the power rule and multiply exponents, .
Step 4.3.19.1.3
Combine and .
Step 4.3.19.1.4
Cancel the common factor of .
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Step 4.3.19.1.4.1
Cancel the common factor.
Step 4.3.19.1.4.2
Rewrite the expression.
Step 4.3.19.1.5
Simplify.
Step 4.3.19.2
Expand using the FOIL Method.
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Step 4.3.19.2.1
Apply the distributive property.
Step 4.3.19.2.2
Apply the distributive property.
Step 4.3.19.2.3
Apply the distributive property.
Step 4.3.19.3
Simplify and combine like terms.
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Step 4.3.19.3.1
Simplify each term.
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Step 4.3.19.3.1.1
Multiply by .
Step 4.3.19.3.1.2
Multiply by .
Step 4.3.19.3.1.3
Multiply by .
Step 4.3.19.3.1.4
Rewrite using the commutative property of multiplication.
Step 4.3.19.3.1.5
Multiply by by adding the exponents.
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Step 4.3.19.3.1.5.1
Move .
Step 4.3.19.3.1.5.2
Multiply by .
Step 4.3.19.3.2
Add and .
Step 4.3.19.3.3
Add and .
Step 4.3.19.4
Rewrite as .
Step 4.3.19.5
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.3.20
Reduce the expression by cancelling the common factors.
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Step 4.3.20.1
Cancel the common factor.
Step 4.3.20.2
Rewrite the expression.
Step 4.3.20.3
Cancel the common factor.
Step 4.3.20.4
Divide by .
Step 4.4
Since and , then is the inverse of .