Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Cancel the common factor of and .

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Divide by .

Add to both sides of the equation.

Simplify the left side of the equation.

Simplify terms.

Simplify each term.

Factor out of .

Apply the sine double–angle identity.

Multiply by .

Use the double–angle identity to transform to .

Apply the distributive property.

Multiply by by adding the exponents.

Move .

Multiply by .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Multiply by by adding the exponents.

Move .

Multiply by .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Multiply by .

Apply the sine double–angle identity.

Multiply by .

Simplify with factoring out.

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Reorder and .

Apply pythagorean identity.

Add and .

Multiply by by adding the exponents.

Move .

Multiply by .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Multiply by .

Divide each term in by .

Cancel the common factor.

Divide by .

Divide by .

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 .

Take the inverse sine of both sides of the equation to extract from inside the sine.

The exact value of is .

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.

Subtract from .

Find the period.

The period of the function can be calculated using .

Replace with in the formula for period.

Solve the equation.

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 .

Take the root of both sides of the to eliminate the exponent on the left side.

Simplify .

Rewrite as .

Pull terms out from under the radical, assuming positive real numbers.

Take the inverse cosine of both sides of the equation to extract from inside the cosine.

The exact value of is .

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from to find the solution in the fourth quadrant.

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.

Multiply by .

Subtract from .

Find the period.

The period of the function can be calculated using .

Replace with in the formula for period.

Solve the equation.

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

Consolidate the answers.

, for any integer

Solve for x 4sin(4x)=-8sin(2x)