Solve for x tan(2x)tan(x)=1

Math
Divide each term by and simplify.
Tap for more steps…
Divide each term in by .
Cancel the common factor of .
Tap for more steps…
Cancel the common factor.
Divide by .
Simplify .
Tap for more steps…
Rewrite in terms of sines and cosines.
Multiply by the reciprocal of the fraction to divide by .
Convert from to .
Move all terms containing to the left side of the equation.
Tap for more steps…
Subtract from both sides of the equation.
Simplify each term.
Tap for more steps…
Apply the tangent doubleangle identity.
Simplify the denominator.
Tap for more steps…
Rewrite as .
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Substitute for .
Find the LCD of the terms in the equation.
Tap for more steps…
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
The LCM is the smallest number that all of the numbers divide into evenly.
1. List the prime factors of each number.
2. Multiply each factor the greatest number of times it occurs in either number.
The number is not a prime number because it only has one positive factor, which is itself.
Not prime
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either number.
The factor for is itself.
occurs time.
The factor for is itself.
occurs time.
The LCM of is the result of multiplying all factors the greatest number of times they occur in either term.
Multiply each term by and simplify.
Tap for more steps…
Multiply each term in by in order to remove all the denominators from the equation.
Simplify each term.
Tap for more steps…
Cancel the common factor of .
Tap for more steps…
Cancel the common factor.
Rewrite the expression.
Expand using the FOIL Method.
Tap for more steps…
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
Tap for more steps…
Simplify each term.
Tap for more steps…
Multiply by .
Multiply by .
Multiply by .
Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
Tap for more steps…
Move .
Multiply by .
Add and .
Add and .
Apply the distributive property.
Multiply by .
Rewrite using the commutative property of multiplication.
Simplify each term.
Tap for more steps…
Multiply by .
Multiply by .
Simplify .
Tap for more steps…
Expand using the FOIL Method.
Tap for more steps…
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
Tap for more steps…
Simplify each term.
Tap for more steps…
Multiply by .
Multiply by .
Multiply by .
Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
Tap for more steps…
Move .
Multiply by .
Add and .
Add and .
Multiply by .
Use the quadratic formula to find the solutions.
Substitute the values , , and into the quadratic formula and solve for .
Simplify.
Tap for more steps…
Simplify the numerator.
Tap for more steps…
Raise to the power of .
Multiply by by adding the exponents.
Tap for more steps…
Move .
Multiply by .
Move to the left of .
Rewrite as .
Multiply by .
Factor out of .
Tap for more steps…
Factor out of .
Factor out of .
Rewrite as .
Tap for more steps…
Rewrite as .
Rewrite as .
Pull terms out from under the radical.
One to any power is one.
Simplify .
Factor out of .
Multiply by .
Multiply by .
The final answer is the combination of both solutions.
Substitute for .
Set up each of the solutions to solve for .
Set up the equation to solve for .
Solve the equation for .
Tap for more steps…
Simplify the numerator.
Tap for more steps…
Apply pythagorean identity.
Pull terms out from under the radical, assuming positive real numbers.
Rewrite the equation as .
Solve for .
Tap for more steps…
Multiply each term by and simplify.
Tap for more steps…
Multiply each term in by .
Simplify the left side of the equation by cancelling the common factors.
Tap for more steps…
Cancel the common factor of .
Tap for more steps…
Move the leading negative in into the numerator.
Cancel the common factor.
Rewrite the expression.
Apply the distributive property.
Multiply by .
Rewrite the equation as .
Divide each term by and simplify.
Tap for more steps…
Divide each term in by .
Cancel the common factor of .
Tap for more steps…
Cancel the common factor.
Divide by .
Move the negative in front of the fraction.
Simplify each term.
Tap for more steps…
Rewrite in terms of sines and cosines.
Multiply by the reciprocal of the fraction to divide by .
Convert from to .
Rewrite in terms of sines and cosines.
Rewrite in terms of sines and cosines.
Multiply by the reciprocal of the fraction to divide by .
Simplify.
Tap for more steps…
Convert from to .
Convert from to .
Since is on the right side of the equation, switch the sides so it is on the left side of the equation.
Factor out of .
Tap for more steps…
Factor out of .
Factor out of .
Factor out of .
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Set the first factor equal to and solve.
Tap for more steps…
Set the first factor equal to .
Take the inverse cotangent of both sides of the equation to extract from inside the cotangent.
The exact value of is .
The cotangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from to find the solution in the fourth quadrant.
Simplify .
Tap for more steps…
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 .
Tap for more steps…
Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Tap for more steps…
Move to the left of .
Add and .
Find the period.
Tap for more steps…
The period of the function can be calculated using .
Replace with in the formula for period.
Solve the equation.
Tap for more steps…
The absolute value is the distance between a number and zero. The distance between and is .
Divide by .
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Set the next factor equal to and solve.
Tap for more steps…
Set the next factor equal to .
Add to both sides of the equation.
Take the inverse cosecant of both sides of the equation to extract from inside the cosecant.
The exact value of is .
The cosecant 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 .
Tap for more steps…
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 .
Tap for more steps…
Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Tap for more steps…
Move to the left of .
Subtract from .
Find the period.
Tap for more steps…
The period of the function can be calculated using .
Replace with in the formula for period.
Solve the equation.
Tap for more steps…
The absolute value is the distance between a number and zero. The distance between and is .
Divide by .
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
The final solution is all the values that make true.
, for any integer
, for any integer
Set up the equation to solve for .
Solve the equation for .
Tap for more steps…
Simplify the numerator.
Tap for more steps…
Rearrange terms.
Apply pythagorean identity.
Pull terms out from under the radical, assuming positive real numbers.
Rewrite the equation as .
Solve for .
Tap for more steps…
Multiply each term by and simplify.
Tap for more steps…
Multiply each term in by .
Simplify the left side of the equation by cancelling the common factors.
Tap for more steps…
Cancel the common factor of .
Tap for more steps…
Move the leading negative in into the numerator.
Cancel the common factor.
Rewrite the expression.
Apply the distributive property.
Multiply by .
Rewrite the equation as .
Divide each term by and simplify.
Tap for more steps…
Divide each term in by .
Cancel the common factor of .
Tap for more steps…
Cancel the common factor.
Divide by .
Simplify each term.
Tap for more steps…
Move the negative in front of the fraction.
Move the negative in front of the fraction.
Simplify each term.
Tap for more steps…
Rewrite in terms of sines and cosines.
Multiply by the reciprocal of the fraction to divide by .
Convert from to .
Rewrite in terms of sines and cosines.
Rewrite in terms of sines and cosines.
Multiply by the reciprocal of the fraction to divide by .
Simplify.
Tap for more steps…
Convert from to .
Convert from to .
Since is on the right side of the equation, switch the sides so it is on the left side of the equation.
Factor out of .
Tap for more steps…
Factor out of .
Factor out of .
Factor out of .
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Set the first factor equal to and solve.
Tap for more steps…
Set the first factor equal to .
Take the inverse cotangent of both sides of the equation to extract from inside the cotangent.
The exact value of is .
The cotangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from to find the solution in the fourth quadrant.
Simplify .
Tap for more steps…
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 .
Tap for more steps…
Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Tap for more steps…
Move to the left of .
Add and .
Find the period.
Tap for more steps…
The period of the function can be calculated using .
Replace with in the formula for period.
Solve the equation.
Tap for more steps…
The absolute value is the distance between a number and zero. The distance between and is .
Divide by .
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Set the next factor equal to and solve.
Tap for more steps…
Set the next factor equal to .
Add to both sides of the equation.
Multiply each term in by
Tap for more steps…
Multiply each term in by .
Multiply .
Tap for more steps…
Multiply by .
Multiply by .
Multiply by .
Take the inverse cosecant of both sides of the equation to extract from inside the cosecant.
The exact value of is .
The cosecant function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from , to find a reference angle. Next, add this reference angle to to find the solution in the third quadrant.
Simplify the expression to find the second solution.
Tap for more steps…
Simplify .
Tap for more steps…
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 .
Tap for more steps…
Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Tap for more steps…
Multiply by .
Add 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 .
Tap for more steps…
Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Tap for more steps…
Move to the left of .
Add and .
Subtract from .
The resulting angle of is positive, less than , and coterminal with .
Find the period.
Tap for more steps…
The period of the function can be calculated using .
Replace with in the formula for period.
Solve the equation.
Tap for more steps…
The absolute value is the distance between a number and zero. The distance between and is .
Divide by .
Add to every negative angle to get positive angles.
Tap for more steps…
Add to to find the positive angle.
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 .
Tap for more steps…
Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Tap for more steps…
Multiply by .
Subtract from .
List the new angles.
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
The final solution is all the values that make true.
, 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 the answers.
Tap for more steps…
Consolidate and to .
, for any integer
Consolidate and to .
, for any integer
Consolidate and to .
, for any integer
Consolidate and to .
, for any integer
Consolidate the answers.
, for any integer
, for any integer
Verify each of the solutions by substituting them into and solving.
, for any integer
Solve for x tan(2x)tan(x)=1

Do you need help with math?

Try our mobile app

Our app allows students to get instant step-by-step solutions to all kinds of math troubles.

Our MATH EXPERTS

JackHudson

Charlie Trom

Lucy Evel

Scroll to top