Trig Identities Question: solve cos squared times x = cos x plus 2?

Answer 1

Do you mean cosine squared of x.
cosine squared by itself is not defined. You have to be taking the cosine of something.

I'll assume the problem is
(cos x)^2 = cos(x) + 2
Move everything to the same side by subtracting cos(x) and 2 from each side
(cos x)^2 - cos(x) - 2 = 0
This is similar to the equation y^2 -y -2 = 0. You remember how to factor equations like that, right?
Factoring the cosine equation, you get
(cos(x) -2)(cos(x) +1) = 0
cos(x) - 2 = 0 or cos(x) + 1 = 0
cos(x) = 2 or cos(x) = -1
If you're familiar with the properties of cosine (and you should be), then you know that cos(x) is never greater than one; therefore, cos(x) = 2 is not true for any x.
cos(x) = -1 when x = pi
In general, x = (2n+1)pi, for n = 0, 1, 2, 3, ...

Answer 2

cos ² x - cos x - 2 = 0
(cos x - 2) (cos x + 1) = 0
cos x = - 1
x = 180°