8) Given: \( \csc \theta=\frac{25}{7} \), find \( \sec \)

To find \( \sec \theta \), we start by recalling the identity relating cosecant and sine. Since \( \csc \theta = \frac{1}{\sin \theta} \), we can find \( \sin \theta \): \[ \sin \theta = \frac{1}{\csc \theta} = \frac{7}{25} \] Next, we can use the Pythagorean identity to find \( \cos \theta \): \[ \sin^2 \theta + \cos^2 \theta = 1 \] \[ \left(\frac{7}{25}\right)^2 + \cos^2 \theta = 1 \] \[ \frac{49}{625} + \cos^2 \theta = 1 \] \[ \cos^2 \theta = 1 - \frac{49}{625} \] \[ \cos^2 \theta = \frac{625 - 49}{625} = \frac{576}{625} \] \[ \cos \theta = \pm \frac{24}{25} \] Finally, since \( \sec \theta = \frac{1}{\cos \theta} \): \[ \sec \theta = \pm \frac{25}{24} \] So, \( \sec \theta = \frac{25}{24} \) or \( -\frac{25}{24} \) depending on the quadrant where \( \theta \) lies.