Question

Find all solutions to the equation. cos^2x + 2 cos x + 1 = 0

Answer 1

x = pi, 3pi, 5pi etc

x = -pi, -3pi, -5pi etc

Explanation:

cos^2 x + 2 cos x + 1 = 0

replace cos x with m

m^2 + 2m +1 = 0

solve by factoring (this is a^2 +2ab+b^2 = (a+b)^2 where a =m and b=1)

(m+1)^2=0

take the square root of each side

m+1 =0

m=-1

now replace m with cos x

cos x = -1

take arccos of each side

arccos cos x = arccos (-1)

x = pi, 3pi, 5pi etc