As Dr. Peterson said, when m = zero, the [imath]\cos(2x)[/imath] term drops out. Mathematically, that is all that needs to be said.
Implicitly, however, we solve this equation using a substitution, namely [imath]\lambda = \cos(x)[/imath]. When m [imath]\ne[/imath] 0, the equation AFTER SUBSTITUTION is quadratic, but is linear if m = 0. The underlying equation, however, is trigonometric whether or not m is zero. As a matter of language, the explanation is confusing because, if you go back to the original equation and let m = 0, you get
[math]- 4\cos(x) - 2 = 0[/math],
an equation that is blatantly not linear. Therefore, the explanation only makes sense if you remember the substitution that was IMPLICIT.
My impression is that, in the teaching of math, there is a lot of implicit substitution that causes confusion for students. In distinction to what your book says, notice how carefully Dr. Peterson talks about “using the quadratic formula.” He never says the equation is linear. He says that if m = 0, the equation is not quadratic, which is certainly true and which is sufficient to justify the fact that the quadratic formula does not apply.