For [tex]180^\circ<\theta<270^\circ[/tex], we expect to have [tex]\cos\theta<0[/tex]. Then if [tex]\sin\theta=-\dfrac{15}{17}[/tex], we have
[tex]cos^2\theta+\sin^2\theta=1\implies\cos\theta=-\sqrt{1-\sin^2\theta}=-\dfrac8{17}[/tex]
[tex]\implies\sec\theta=\boxed{-\dfrac{17}8}[/tex]
