# Solve over the Interval cos(x)^2+2cos(x)+1=0 , [0,2pi] ,
Factor the left side of the equation.
Let . Substitute for all occurrences of .
Factor using the perfect square rule.
Rewrite as .
Check the middle term by multiplying and compare this result with the middle term in the original expression.
Simplify.
Factor using the perfect square trinomial rule , where and .
Replace all occurrences of with .
Replace the left side with the factored expression.
Set equal to and solve for .
Set the factor equal to .
Subtract from both sides of the equation.
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 negative in the second and third quadrants. To find the second solution, subtract the reference angle from to find the solution in the third 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
Plug in for and simplify to see if the solution is contained in .
Plug in for .
Simplify.
Multiply .
Multiply by .
Multiply by .     