Solve sinx+cosx/2=6cosx/35sinxcosx

$$\frac { \sin x + \cos x } { 2 } = \frac { 6 \cos x } { 35 \sin x \cos x }$$

Solve for x (complex solution)

$$x=\frac{1}{2}ArcTanI(\frac{1225}{1104}+\left(-\frac{11}{1104}\right)\times 2329^{\frac{1}{2}})+\pi n_{12}+\left(-\frac{1}{2}i\right)\ln(|\frac{11}{70}-\frac{11}{70}i+\left(\frac{1}{70}+\frac{1}{70}i\right)\times 2329^{\frac{1}{2}}|)\text{, }n_{12}\in \mathrm{Z}$$

$$x=n_{19}\pi +\left(-\frac{1}{2}i\right)\ln(|\frac{11}{70}-\frac{11}{70}i+\left(-\frac{1}{70}-\frac{1}{70}i\right)\times 2329^{\frac{1}{2}}|)+\frac{1}{2}ArcTanI(\frac{1225}{1104}+\frac{11}{1104}\times 2329^{\frac{1}{2}})+\frac{1}{2}\pi \text{, }n_{19}\in \mathrm{Z}$$

Solve for x

$$x=\pi n_{1}+\arcsin(\frac{\sqrt{35\left(59-\sqrt{2329}\right)}}{70})\text{, }n_{1}\in \mathrm{Z}$$

$$x=\pi n_{2}-\arcsin(\frac{\sqrt{35\left(\sqrt{2329}+59\right)}}{70})+\pi \text{, }n_{2}\in \mathrm{Z}$$