Verified answer

cos²x = 2cosx if and only if cosx = 0 so that is not an identity.

Perhaps you are thinking of the double angle identity:

sin(2x) = 2sin(x)cos(x)

or

cos(2x) = cos²x - sin²x = 2cos²x - 1 = 1 - 2sin²x

it's cos²x

If cos(x)cos(x) = 2cos(x) then cos(x) = 2 which is not true obviously, cos(x) is never 2.

cos²x does not equal 2cosx

you confuse it with the derivative of cos²x, which is 2cos x if I'm correct

EDIT: whoops, my bad. the derivative of cos²x isn't 2cos x. sorry.

No! If cos(x)cos(x) = 2cos(x)

then cos(x) = 2, which is clearly not true.

No.