Use the information given about the angle $\theta, \cot \theta=-5, \sec \theta< 0,0 \leq \theta< 2 \pi$, to find the exact values of the following.
(a) $\sin (2 \theta),(b) \cos (2 \theta),(c) \sin \frac{\theta}{2}$, and (d) $\cos \frac{\theta}{2}$

Step 1 ：Given that \(\cot \theta = -5\) and \(\sec \theta < 0\), we can infer that \(\theta\) is in the second or fourth quadrant. However, since \(0 \leq \theta < 2\pi\), \(\theta\) must be in the second quadrant. In the second quadrant, cosine is negative and sine is positive.

Step 2 ：We can use the identity \(\cot^2 \theta + 1 = \csc^2 \theta\) to find \(\csc \theta\), and then find \(\sin \theta\) as the reciprocal of \(\csc \theta\).

Step 3 ：Then, we can use the double angle formulas \(\sin(2\theta) = 2\sin\theta\cos\theta\) and \(\cos(2\theta) = \cos^2\theta - \sin^2\theta\) to find \(\sin(2\theta)\) and \(\cos(2\theta)\).

Step 4 ：Finally, we can use the half angle formulas \(\sin(\theta/2) = \pm \sqrt{(1 - \cos\theta)/2}\) and \(\cos(\theta/2) = \pm \sqrt{(1 + \cos\theta)/2}\) to find \(\sin(\theta/2)\) and \(\cos(\theta/2)\). Since \(\theta\) is in the second quadrant, \(\theta/2\) is in the first quadrant, so both \(\sin(\theta/2)\) and \(\cos(\theta/2)\) are positive.

Step 5 ：Final Answer: \(\boxed{\sin (2 \theta) = -0.3846}\), \(\boxed{\cos (2 \theta) = 0.9231}\), \(\boxed{\sin \frac{\theta}{2} = 0.9951}\), \(\boxed{\cos \frac{\theta}{2} = 0.0985}\)