# Solve for x cos(x)^4-sin(x)^4=cos(2x)

Use the identity to solve the equation. In this identity, represents the angle created by plotting point on a graph and therefore can be found using .
where and
Set up the equation to find the value of .
Take the inverse tangent to solve the equation for .
Divide by .
The exact value of is .
Solve to find the value of .
Raise to the power of .
One to any power is one.
Substitute the known values into the equation.
Multiply by .
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Simplify .
Multiply by .
Combine and simplify the denominator.
Multiply and .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Rewrite as .
Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Evaluate the exponent.
Move all terms containing to the left side of the equation.
Subtract from both sides of the equation.
Simplify the left side of the equation.
Simplify the numerator.
Use the doubleangle identity to transform to .
Since both terms are perfect squares, factor using the difference of squares formula, where and .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Move to the left of .
Apply the distributive property.
Expand using the FOIL Method.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
Simplify each term.
Multiply .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Multiply .
Multiply by .
Multiply by .
Multiply .
Multiply by .
Multiply by .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Reorder the factors of .
Subtract from .
Apply the distributive property.
Factor out of .
Move .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Set equal to and solve for .
Set the factor equal to .
Simplify the left side.
Apply the distributive property.
Simplify.
Multiply .
Multiply by .
Multiply by .
Multiply by .
Move all terms not containing to the right side of the equation.
Add to both sides of the equation.
Subtract from both sides of the equation.
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
The solution is the result of .
Move to the left side of the equation by subtracting it from both sides.
Simplify the left side.
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Move to the left of .
Apply the distributive property.
Multiply .
Multiply by .
Multiply by .
Factor out of .
Move .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Set equal to and solve for .
Set the factor equal to .
Simplify the left side.
Apply the distributive property.
Simplify.
Multiply .
Multiply by .
Multiply by .
Multiply by .
Move all terms not containing to the right side of the equation.
Add to both sides of the equation.
Subtract from both sides of the equation.
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
The solution is the result of .
Move to the left side of the equation by subtracting it from both sides.
Simplify the left side.
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Move to the left of .
Apply the distributive property.
Multiply .
Multiply by .
Multiply by .
Factor out of .
Move .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Set equal to and solve for .
Set the factor equal to .
Simplify the left side.
Apply the distributive property.
Simplify.
Multiply .
Multiply by .
Multiply by .
Multiply by .
Move all terms not containing to the right side of the equation.
Add to both sides of the equation.
Subtract from both sides of the equation.
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
The solution is the result of .
Since is on the right side of the equation, switch the sides so it is on the left side of the equation.
Move to the left side of the equation by subtracting it from both sides.
Simplify the left side.
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Multiply by .
Combine the numerators over the common denominator.
Multiply by .
Multiply both sides of the equation by .
Remove parentheses.
Multiply by .
Move all terms not containing to the right side of the equation.
Add to both sides of the equation.
Add to both sides of the equation.
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Simplify .
Multiply by .
Combine and simplify the denominator.
Multiply and .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Rewrite as .
Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Evaluate the exponent.
Apply the distributive property.
Multiply .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Simplify each term.
Rewrite as .
Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Evaluate the exponent.
Move to the left of .
Cancel the common factor of and .
Factor out of .
Factor out of .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Take the root of both sides of the to eliminate the exponent on the left side.
The complete solution is the result of both the positive and negative portions of the solution.
First, use the positive value of the to find the first solution.
Next, use the negative value of the to find the second solution.
The complete solution is the result of both the positive and negative portions of the solution.
Set up each of the solutions to solve for .
Set up the equation to solve for .
Solve the equation for .
Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.
To remove the radical on the left side of the equation, square both sides of the equation.
Simplify the left side of the equation.
Solve for .
Move to the left side of the equation by subtracting it from both sides.
Replace with .
Simplify .
Replace the with based on the identity.
Combine the opposite terms in .
Subtract from .
Reorder the polynomial.
Replace with .
Simplify .
Move all terms not containing to the right side of the equation.
Add to both sides of the equation.
Subtract from both sides of the equation.
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Take the root of both sides of the to eliminate the exponent on the left side.
The complete solution is the result of both the positive and negative portions of the solution.
Simplify the right side of the equation.
Rewrite as .
Multiply by .
Combine and simplify the denominator.
Multiply and .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Rewrite as .
Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Evaluate the exponent.
Combine using the product rule for radicals.
The complete solution is the result of both the positive and negative portions of the solution.
First, use the positive value of the to find the first solution.
Next, use the negative value of the to find the second solution.
The complete solution is the result of both the positive and negative portions of the solution.
Set up each of the solutions to solve for .
Set up the equation to solve for .
Solve the equation for .
Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.
Simplify the numerator.
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Multiply by .
Combine the numerators over the common denominator.
Move to the left of .
Multiply both sides of the equation by .
Move to the left of .
Move all terms containing to the left side of the equation.
Subtract from both sides of the equation.
Subtract from .
Multiply each term in by
Multiply each term in by .
Multiply .
Multiply by .
Multiply by .
Multiply by .
To remove the radical on the left side of the equation, square both sides of the equation.
Simplify each side of the equation.
Simplify the left side of the equation.
Raising to any positive power yields .
Solve for .
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Divide by .
Subtract from both sides of the equation.
Divide each term by and simplify.
Divide each term in by .
Simplify the left side of the equation by multiplying out all the terms.
Dividing two negative values results in a positive value.
Cancel the common factor of .
Cancel the common factor.
Divide by .
Simplify .
Dividing two negative values results in a positive value.
Multiply by .
Combine and simplify the denominator.
Multiply and .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Rewrite as .
Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Evaluate the exponent.
Take the inverse sine of both sides of the equation to extract from inside the sine.
Simplify the left side.
Split the fraction into two fractions.
Simplify each term.
Cancel the common factor of .
Cancel the common factor.
Divide by .
Move the negative in front of the fraction.
The exact value of is .
Move all terms not containing to the right side of the equation.
Add to both sides of the equation.
Simplify the right side of the equation.
Combine the numerators over the common denominator.
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from to find the solution in the second quadrant.
Simplify the expression to find the second solution.
Simplify .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Move to the left of .
Subtract from .
Move all terms not containing to the right side of the equation.
Add to both sides of the equation.
Simplify the right side of the equation.
Combine the numerators over the common denominator.
Cancel the common factor of .
Cancel the common factor.
Divide by .
Find the period.
The period of the function can be calculated using .
Replace with in the formula for period.
Solve the equation.
is approximately which is positive so remove the absolute value
Multiply the numerator by the reciprocal of the denominator.
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply by .
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Set up the equation to solve for .
Solve the equation for .
Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.
Simplify the numerator.
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Multiply by .
Combine the numerators over the common denominator.
Move to the left of .
Multiply both sides of the equation by .
Simplify both sides of the equation.
Multiply .
Multiply by .
Multiply by .
Rewrite as .
Multiply both sides of the equation by .
Multiply by .
Move all terms containing to the left side of the equation.
Add to both sides of the equation.
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Divide by .
To remove the radical on the left side of the equation, square both sides of the equation.
Simplify each side of the equation.
Simplify the left side of the equation.
Raising to any positive power yields .
Solve for .
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Divide by .
Subtract from both sides of the equation.
Divide each term by and simplify.
Divide each term in by .
Simplify the left side of the equation by multiplying out all the terms.
Dividing two negative values results in a positive value.
Cancel the common factor of .
Cancel the common factor.
Divide by .
Simplify .
Dividing two negative values results in a positive value.
Multiply by .
Combine and simplify the denominator.
Multiply and .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Rewrite as .
Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Evaluate the exponent.
Take the inverse sine of both sides of the equation to extract from inside the sine.
Simplify the left side.
Split the fraction into two fractions.
Simplify each term.
Cancel the common factor of .
Cancel the common factor.
Divide by .
Move the negative in front of the fraction.
The exact value of is .
Move all terms not containing to the right side of the equation.
Add to both sides of the equation.
Simplify the right side of the equation.
Combine the numerators over the common denominator.
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from to find the solution in the second quadrant.
Simplify the expression to find the second solution.
Simplify .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Move to the left of .
Subtract from .
Move all terms not containing to the right side of the equation.
Add to both sides of the equation.
Simplify the right side of the equation.
Combine the numerators over the common denominator.
Cancel the common factor of .
Cancel the common factor.
Divide by .
Find the period.
The period of the function can be calculated using .
Replace with in the formula for period.
Solve the equation.
is approximately which is positive so remove the absolute value
Multiply the numerator by the reciprocal of the denominator.
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply by .
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
List all of the results found in the previous steps.
, for any integer
The complete solution is the set of all solutions.
, for any integer
, for any integer
, for any integer
Set up the equation to solve for .
Solve the equation for .
Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.
To remove the radical on the left side of the equation, square both sides of the equation.
Simplify the left side of the equation.
Solve for .
Move to the left side of the equation by subtracting it from both sides.
Simplify each term.
Raise to the power of .
Multiply by .
Replace with .
Simplify .
Replace the with based on the identity.
Combine the opposite terms in .
Subtract from .
Reorder the polynomial.
Replace with .
Simplify .
Move all terms not containing to the right side of the equation.
Add to both sides of the equation.
Subtract from both sides of the equation.
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Take the root of both sides of the to eliminate the exponent on the left side.
The complete solution is the result of both the positive and negative portions of the solution.
Simplify the right side of the equation.
Rewrite as .
Multiply by .
Combine and simplify the denominator.
Multiply and .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Rewrite as .
Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Evaluate the exponent.
Combine using the product rule for radicals.
The complete solution is the result of both the positive and negative portions of the solution.
First, use the positive value of the to find the first solution.
Next, use the negative value of the to find the second solution.
The complete solution is the result of both the positive and negative portions of the solution.
Set up each of the solutions to solve for .
Set up the equation to solve for .
Solve the equation for .
Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.
Simplify the numerator.
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Multiply by .
Combine the numerators over the common denominator.
Move to the left of .
Multiply both sides of the equation by .
Move to the left of .
Move all terms containing to the left side of the equation.
Subtract from both sides of the equation.
Subtract from .
Multiply each term in by
Multiply each term in by .
Multiply .
Multiply by .
Multiply by .
Multiply by .
To remove the radical on the left side of the equation, square both sides of the equation.
Simplify each side of the equation.
Simplify the left side of the equation.
Raising to any positive power yields .
Solve for .
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Divide by .
Subtract from both sides of the equation.
Divide each term by and simplify.
Divide each term in by .
Simplify the left side of the equation by multiplying out all the terms.
Dividing two negative values results in a positive value.
Cancel the common factor of .
Cancel the common factor.
Divide by .
Simplify .
Dividing two negative values results in a positive value.
Multiply by .
Combine and simplify the denominator.
Multiply and .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Rewrite as .
Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Evaluate the exponent.
Take the inverse sine of both sides of the equation to extract from inside the sine.
Simplify the left side.
Split the fraction into two fractions.
Simplify each term.
Cancel the common factor of .
Cancel the common factor.
Divide by .
Move the negative in front of the fraction.
The exact value of is .
Move all terms not containing to the right side of the equation.
Add to both sides of the equation.
Simplify the right side of the equation.
Combine the numerators over the common denominator.
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from to find the solution in the second quadrant.
Simplify the expression to find the second solution.
Simplify .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Move to the left of .
Subtract from .
Move all terms not containing to the right side of the equation.
Add to both sides of the equation.
Simplify the right side of the equation.
Combine the numerators over the common denominator.
Cancel the common factor of .
Cancel the common factor.
Divide by .
Find the period.
The period of the function can be calculated using .
Replace with in the formula for period.
Solve the equation.
is approximately which is positive so remove the absolute value
Multiply the numerator by the reciprocal of the denominator.
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply by .
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Set up the equation to solve for .
Solve the equation for .
Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.
Simplify the numerator.
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Multiply by .
Combine the numerators over the common denominator.
Move to the left of .
Multiply both sides of the equation by .
Simplify both sides of the equation.
Multiply .
Multiply by .
Multiply by .
Rewrite as .
Multiply both sides of the equation by .
Multiply by .
Move all terms containing to the left side of the equation.
Add to both sides of the equation.
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Divide by .
To remove the radical on the left side of the equation, square both sides of the equation.
Simplify each side of the equation.
Simplify the left side of the equation.
Raising to any positive power yields .
Solve for .
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Divide by .
Subtract from both sides of the equation.
Divide each term by and simplify.
Divide each term in by .
Simplify the left side of the equation by multiplying out all the terms.
Dividing two negative values results in a positive value.
Cancel the common factor of .
Cancel the common factor.
Divide by .
Simplify .
Dividing two negative values results in a positive value.
Multiply by .
Combine and simplify the denominator.
Multiply and .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Rewrite as .
Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Evaluate the exponent.
Take the inverse sine of both sides of the equation to extract from inside the sine.
Simplify the left side.
Split the fraction into two fractions.
Simplify each term.
Cancel the common factor of .
Cancel the common factor.
Divide by .
Move the negative in front of the fraction.
The exact value of is .
Move all terms not containing to the right side of the equation.
Add to both sides of the equation.
Simplify the right side of the equation.
Combine the numerators over the common denominator.
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from to find the solution in the second quadrant.
Simplify the expression to find the second solution.
Simplify .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Move to the left of .
Subtract from .
Move all terms not containing to the right side of the equation.
Add to both sides of the equation.
Simplify the right side of the equation.
Combine the numerators over the common denominator.
Cancel the common factor of .
Cancel the common factor.
Divide by .
Find the period.
The period of the function can be calculated using .
Replace with in the formula for period.
Solve the equation.
is approximately which is positive so remove the absolute value
Multiply the numerator by the reciprocal of the denominator.
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply by .
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
List all of the results found in the previous steps.
, for any integer
The complete solution is the set of all solutions.
, for any integer
, for any integer
, for any integer
List all of the results found in the previous steps.
, for any integer
The complete solution is the set of all solutions.
, for any integer
Consolidate and to .
, for any integer
Consolidate and to .
, for any integer
Consolidate and to .
, for any integer
Consolidate and to .
, for any integer
, for any integer
Solve for x cos(x)^4-sin(x)^4=cos(2x)