Solve S=1/2(32sin2θ-1/tanθ)

$$S = \frac { 1 } { 2 } ( 32 \sin 2 \theta - \frac { 1 } { \tan \theta } )$$

Solve for S (complex solution)

$$\left\{\begin{matrix}S=\frac{i\left(16i\cos(3\theta )-15i\cos(\theta )\right)}{2\sin(\theta )}\text{, }&\nexists n_{2}\in \mathrm{Z}\text{ : }\theta =\frac{\pi n_{2}}{2}\\S\in \mathrm{C}\text{, }&\nexists n_{2}\in \mathrm{Z}\text{ : }\theta =\frac{\pi n_{2}}{2}\text{ and }2\left(16i\cos(3\theta )-15i\cos(\theta )\right)=0\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}\end{matrix}\right.$$

Solve for S

$$S=\frac{32\sin(2\theta )-\cot(\theta )}{2}$$

$$\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =\frac{\pi n_{1}}{2}$$