Answer
(a) $ -\frac{\sqrt 3}{2}$
(b) $ -\frac{1}{2}$
(c) $ \frac{1}{2}$
(d) $ -\frac{\sqrt 3}{2}$
Work Step by Step
Given $sec\theta=2, csc\theta\lt0$, we know $\frac{3\pi}{2}\lt \theta\lt 2\pi$. We have $cos\theta=\frac{1}{2},sin\theta=-\frac{\sqrt 3}{2}$, and $\frac{3\pi}{4}\lt \frac{\theta}{2} \lt \pi$.
(a) $sin(2\theta)=2sin\theta cos\theta=2(-\frac{\sqrt 3}{2})(\frac{1}{2})=-\frac{\sqrt 3}{2}$
(b) $cos(2\theta)=1-2sin^2(\theta)=1-2(-\frac{\sqrt 3}{2})^2=-\frac{1}{2}$
(c) $sin(\frac{\theta}{2})=\sqrt {\frac{1-cos\theta}{2}}=\sqrt {\frac{1-1/2}{2}}=\frac{1}{2}$
(d) $cos(\frac{\theta}{2})=-\sqrt {\frac{1+cos\theta}{2}}=-\sqrt {\frac{1+1/2}{2}}=-\frac{\sqrt 3}{2}$
