Given the statements \(\tan \theta = -\frac{12}{5}\) and \(\csc \theta = -\frac{13}{12}\), and the terminal point determined by \(\theta\) is in quadrant 3, let's evaluate each option to determine the correct answer:
1. Option A: cannot be true because \(\tan \theta\) must be less than 1.
This is incorrect. The value of \(\tan \theta\) can be any real number, positive or negative, and is not restricted to being less than 1.
2. Option B: cannot be true because \(12^2 + 5^2 \neq 1\).
To verify, we note that \(12^2 + 5^2 = 144 + 25 = 169\), which indeed does not equal 1. However, this option is a misunderstanding because it is not necessary for the squares of the coefficients in the tangent function to sum up to 1. Instead, they represent the ratio, which can be reduced give the accurate representation of the trigonometric function.
3. Option C: cannot be true because if \(\tan \theta = -\frac{12}{5}\), then \(\csc \theta = \pm \frac{13}{5}\).
This option is also incorrect. The cosecant function is the reciprocal of the sine, and it is not derived simply from the tangent. Thus, \(\csc \theta\) should be calculated separately based on the given conditions, and the condition as given (\(\csc \theta = -\frac{13}{12}\)) does satisfy the trigonometric identities.
4. Option D: cannot be true because \(\tan \theta\) is greater than zero in quadrant 3.
Recall that in quadrant 3, both sine and cosine are negative, making the tangent (which is sine divided by cosine) positive. Given that \(\tan \theta = -\frac{12}{5}\) is negative, it cannot be true in quadrant 3 since tangent where both values would be negative would result in positive tan:
Therefore, the correct answer is:
D. cannot be true because [tex]\(\tan \theta\)[/tex] is greater than zero in quadrant 3.
