# Find the Inverse sec(arctan(x/3))

Interchange the variables.
Solve for .
Rewrite the equation as .
Multiply both sides of the equation by .
Move to the left of .
If is a positive integer that is greater than and is a real number or a factor, then .
Raise each side of the equation to the power to eliminate the fractional exponent on the left side.
Solve the equation for .
Simplify .
Apply the product rule to .
Raise to the power of .
Subtract from both sides of the equation.
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.
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Rewrite as .
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Rewrite as .
Rewrite as .
Rewrite as .
Pull terms out from under the radical.
One to any power is one.
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.
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 .
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 .
Divide by .
One to any power is one.
Apply the product rule to .
Raise to the power of .
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 .
Rewrite as .
Factor the perfect power out of .
Factor the perfect power out of .
Rearrange the fraction .
Pull terms out from under the radical.
Combine 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.
Multiply by .
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 .
Divide by .
One to any power is one.
Apply the product rule to .
Raise to the power of .
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 .
Rewrite as .
Factor the perfect power out of .
Factor the perfect power out of .
Rearrange the fraction .
Pull terms out from under the radical.
Combine 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 .
Simplify terms.
Combine the numerators over the common denominator.
Multiply by .
Multiply and .
Multiply by .
Expand using the FOIL Method.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify terms.
Combine the opposite terms in .
Reorder the factors in the terms and .
Simplify each term.
Multiply .
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 .
Simplify.
Multiply by .
Combine the opposite terms in .
Subtract from .
Rewrite as .
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Since , is the inverse of .
Find the Inverse sec(arctan(x/3))
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