Solve sin^{2}θ+cos^{2}θ=(a/sqrt{4b^{2-3a^{2}}})^{2}+(frac{2sqrt{b^{2}-a^2}}{sqrt{4b^2-3a^2}})^2

$$\sin ^ { 2 } \theta + \cos ^ { 2 } \theta = ( \frac { a } { \sqrt { 4 b ^ { 2 } - 3 a ^ { 2 } } } ) ^ { 2 } + ( \frac { 2 \sqrt { b ^ { 2 } - a ^ { 2 } } } { \sqrt { 4 b ^ { 2 } - 3 a ^ { 2 } } } ) ^ { 2 }$$

Solve for a (complex solution)

$$a\in \mathrm{C}\setminus -\frac{2\sqrt{3}b}{3},\frac{2\sqrt{3}b}{3}$$

Solve for b (complex solution)

$$b\in \mathrm{C}\setminus \frac{\sqrt{3}a}{2},-\frac{\sqrt{3}a}{2}$$

Solve for a

$$\left\{\begin{matrix}a\in \mathrm{R}\setminus ((-\infty,-b)\cup (b,\infty)\cup (-\infty,-\frac{2\sqrt{3}b}{3}]\cup [\frac{2\sqrt{3}b}{3},\infty))\text{, }&b>0\\a\in \mathrm{R}\setminus ((-\infty,b)\cup (-b,\infty)\cup (-\infty,\frac{2\sqrt{3}b}{3}]\cup [-\frac{2\sqrt{3}b}{3},\infty))\text{, }&b<0\end{matrix}\right.$$

Solve for b

$$\left\{\begin{matrix}b\in \mathrm{R}\setminus \left(-\frac{\sqrt{3}a}{2},\frac{\sqrt{3}a}{2}\cup \left(-\frac{\sqrt{3}|a|}{2},\frac{\sqrt{3}|a|}{2}\right)\text{, }a\neq 0\cup \left(-|a|,|a|\right)\text{, }a\neq 0\right)\text{, }&a\neq 0\\b\neq 0\text{, }&a=0\end{matrix}\right.$$