Let's evaluate each statement step-by-step:
### Statement A:
"A cannot be true because [tex]\(\tan \theta\)[/tex] is greater than zero in quadrant 3."
- In quadrant III, the tangent function is positive because both sine and cosine are negative, and [tex]\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)[/tex]. However, here we are given [tex]\(\tan \theta = -\frac{12}{5}\)[/tex], which is negative. Therefore, statement A is true because [tex]\(\tan \theta\)[/tex] should be positive in quadrant 3, but it is given as negative.
### Statement B:
"B cannot be true because [tex]\(12^2 - 5^2 \neq 1\)[/tex]."
- Let's verify:
[tex]\(12^2 - 5^2 = 144 - 25 = 119\)[/tex].
Therefore, [tex]\(12^2 - 5^2 \neq 1\)[/tex].
Hence, statement B is true.
### Statement C:
"C cannot be true because if [tex]\(\tan \theta = -\frac{12}{5}\)[/tex], then [tex]\(\csc \theta = \pm \frac{13}{5}\)[/tex]."
- First, let's confirm if the given [tex]\(\tan \theta = \frac{\sin \theta}{\cos \theta} = -\frac{12}{5}\)[/tex].
If [tex]\(\sin \theta = -\frac{12}{13}\)[/tex] and [tex]\(\cos \theta = \frac{5}{13}\)[/tex], then:
[tex]\(\csc \theta = \frac{1}{\sin \theta} = -\frac{13}{12}\)[/tex].
Given [tex]\(\csc \theta = -\frac{13}{12}\)[/tex], it matches.
Thus, statement C is false since [tex]\(\csc \theta = -\frac{13}{12}\)[/tex], not [tex]\(\pm \frac{13}{5}\)[/tex].
### Statement D:
"D cannot be true because [tex]\(\tan \theta\)[/tex] must be less than 1."
- Here, [tex]\(\tan \theta = -\frac{12}{5}\)[/tex].
The absolute value is:
[tex]\(\left| \tan \theta \right| = \left| -\frac{12}{5} \right| = \frac{12}{5} > 1\)[/tex].
Statement D is, therefore, true because [tex]\(\left| \tan \theta \right| > 1\)[/tex].
Therefore, the statements that cannot be true are A, B, C, and D. The correct answer is:
[tex]\[
(1, 2, 3, 4)
\][/tex]
