Find all solutions to the equation. cos²x + 2 cos x + 1 = 0

Solution:

Given, cos(2x) + 2 cos(x) + 1 = 0

We have to find the solution to the equation.

Using trigonometric identities,

cos(2x) = cos²x - sin²x

cos²x + sin²x = 1

So, cos²x = 1 - sin²x

cos²x - sin²x + 2 cos(x) + 1 = 0

On rearranging,

cos²x + 2 cos(x) + 1 - sin²x = 0

cos²x + cos²x + 2 cos(x) = 0

2cos²x + 2 cos(x) = 0

Dividing by 2 on both sides,

cos²x + cos(x) = 0

Taking out common term,

cos(x)[cos(x) + 1] = 0

So, cos(x) = 0

Taking inverse,

x = cos^-1(0)

x = 𝜋/2, 𝜋/3,...etc

cos(x) + 1 = 0

cos(x) = -1

Taking inverse,

x = cos^-1(-1)

x = 𝜋, 3𝜋,...etc

Therefore, the solutions are 𝜋/2, 𝜋/3, 𝜋, 3𝜋,...etc.