Solve y=cos4θ-4sin^2θ

$$y = \cos 4 \theta - 4 \sin ^ { 2 } \theta$$

Solve for y (complex solution)

$$y=\cos(4\theta )+2\cos(2\theta )-2$$

Solve for y

$$y=8\left(\cos(\theta )\right)^{4}-4\left(\cos(\theta )\right)^{2}-3$$

Solve for θ

$$\theta =\left(-1\right)\left(\left(-2\right)n_{4}\pi +ArcCosI(\frac{1}{2}\left(1+\left(-1\right)\left(7+2y\right)^{\frac{1}{2}}\right)^{\frac{1}{2}})\right)\text{, }n_{4}\in \mathrm{Z}$$

$$\theta =\pi +\left(-1\right)ArcCosI(\frac{1}{2}\left(1+\left(-1\right)\left(7+2y\right)^{\frac{1}{2}}\right)^{\frac{1}{2}})+2\pi n_{38}\text{, }n_{38}\in \mathrm{Z}$$

$$\theta =2\pi n_{39}+\left(-1\right)\pi +ArcCosI(\frac{1}{2}\left(1+\left(-1\right)\left(7+2y\right)^{\frac{1}{2}}\right)^{\frac{1}{2}})\text{, }n_{39}\in \mathrm{Z}$$

$$\theta =ArcCosI(\frac{1}{2}\left(1+\left(-1\right)\left(7+2y\right)^{\frac{1}{2}}\right)^{\frac{1}{2}})+2n_{3}\pi \text{, }n_{3}\in \mathrm{Z}$$

$$\theta =\left(-1\right)\left(\left(-2\right)n_{52}\pi +ArcCosI(\frac{1}{2}\left(1+\left(7+2y\right)^{\frac{1}{2}}\right)^{\frac{1}{2}})\right)\text{, }n_{52}\in \mathrm{Z}$$

$$\theta =\pi +\left(-1\right)ArcCosI(\frac{1}{2}\left(1+\left(7+2y\right)^{\frac{1}{2}}\right)^{\frac{1}{2}})+2\pi n_{82}\text{, }n_{82}\in \mathrm{Z}$$

$$\theta =2\pi n_{83}+\left(-1\right)\pi +ArcCosI(\frac{1}{2}\left(1+\left(7+2y\right)^{\frac{1}{2}}\right)^{\frac{1}{2}})\text{, }n_{83}\in \mathrm{Z}$$

$$\theta =ArcCosI(\frac{1}{2}\left(1+\left(7+2y\right)^{\frac{1}{2}}\right)^{\frac{1}{2}})+2n_{51}\pi \text{, }n_{51}\in \mathrm{Z}$$