# Trigonometry Examples

Factor out of .

Factor out of .

Reorder and .

Factor out of .

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor is 2nd order, terms are required in the numerator. The number of terms required in the numerator is always equal to the order of the factor in the denominator.

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is .

Remove parentheses around .

Reduce the expression by cancelling the common factors.

Cancel the common factor.

Rewrite the expression.

Reduce the expression by cancelling the common factors.

Cancel the common factor.

Divide by to get .

Simplify each term.

Remove parentheses around .

Reduce the expression by cancelling the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Divide by to get .

Apply the distributive property.

Move to the left of the expression .

Multiply by to get .

Remove parentheses around .

Reduce the expression by cancelling the common factors.

Cancel the common factor.

Divide by to get .

Apply the distributive property.

Move .

Use the power rule to combine exponents.

Add and to get .

Remove unnecessary parentheses.

Create an equation for the partial fraction variables by equating the coefficients of from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

Create an equation for the partial fraction variables by equating the coefficients of from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

Create an equation for the partial fraction variables by equating the coefficients of the terms not containing . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

Set up the system of equations to find the coefficients of the partial fractions.

Solve for in the first equation.

Solve for in the second equation.

Replace all occurrences of with the solution found by solving the last equation for . In this case, the value substituted is .

Remove parentheses around .

Solve for in the third equation.

Replace each of the partial fraction coefficients in with the values found for , , and .

Remove parentheses.

Multiply by to get .

Move the negative in front of the fraction.