Find the Inverse cos(4x)-sin(4x)

Interchange the variables.
Solve for .
Rewrite the equation as .
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.
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.
Take the inverse sine of both sides of the equation to extract from inside the sine.
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 .
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 the numerator by the reciprocal of the denominator.
Multiply .
Multiply and .
Multiply by .
Solve for and replace with .
Replace the with to show the final answer.
Set up the composite result function.
Evaluate by substituting in the value of into .
Reorder factors in .
Since , is the inverse of .
Find the Inverse cos(4x)-sin(4x)