The sign of \( \sec \left(-\frac{1}{4} \pi\right) \) is positive negative zero undefined The sign of \( \csc \left(-\frac{1}{4} \pi\right) \) is positive negative zero undefined

To determine the signs of \( \sec \left(-\frac{1}{4} \pi\right) \) and \( \csc \left(-\frac{1}{4} \pi\right) \), we first need to find the values of \( \cos \left(-\frac{1}{4} \pi\right) \) and \( \sin \left(-\frac{1}{4} \pi\right) \). 1. **Finding \( \sec \left(-\frac{1}{4} \pi\right) \)**: \[ \sec \theta = \frac{1}{\cos \theta} \] We first calculate \( \cos \left(-\frac{1}{4} \pi\right) \): \[ \cos \left(-\frac{1}{4} \pi\right) = \cos \left(\frac{1}{4} \pi\right) = \frac{\sqrt{2}}{2} \] Since \( \cos \left(-\frac{1}{4} \pi\right) \) is positive, we have: \[ \sec \left(-\frac{1}{4} \pi\right) = \frac{1}{\cos \left(-\frac{1}{4} \pi\right)} = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \] Therefore, the sign of \( \sec \left(-\frac{1}{4} \pi\right) \) is **positive**. 2. **Finding \( \csc \left(-\frac{1}{4} \pi\right) \)**: \[ \csc \theta = \frac{1}{\sin \theta} \] We calculate \( \sin \left(-\frac{1}{4} \pi\right) \): \[ \sin \left(-\frac{1}{4} \pi\right) = -\sin \left(\frac{1}{4} \pi\right) = -\frac{\sqrt{2}}{2} \] Since \( \sin \left(-\frac{1}{4} \pi\right) \) is negative, we have: \[ \csc \left(-\frac{1}{4} \pi\right) = \frac{1}{\sin \left(-\frac{1}{4} \pi\right)} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}} = -\sqrt{2} \] Therefore, the sign of \( \csc \left(-\frac{1}{4} \pi\right) \) is **negative**. In summary: - The sign of \( \sec \left(-\frac{1}{4} \pi\right) \) is **positive**. - The sign of \( \csc \left(-\frac{1}{4} \pi\right) \) is **negative**.