Solve PQ^2=2-2(cosalphacosbeta+sinalphasinbeta)

$$PQ ^ { 2 } = 2 - 2 ( \cos \alpha \cos \beta + \sin \alpha \sin \beta )$$

Solve for P (complex solution)

$$\left\{\begin{matrix}P=-\frac{\left(\frac{e^{i\alpha -i\beta }-1}{Q}\right)^{2}}{e^{i\alpha -i\beta }}\text{, }&Q\neq 0\\P\in \mathrm{C}\text{, }&\exists n_{1}\in \mathrm{Z}\text{ : }\alpha =2\pi n_{1}+\beta \text{ and }Q=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply -1 and 2 to get -2.

$$PQ^{2}=2-2\left(\cos(\alpha )\cos(\beta )+\sin(\alpha )\sin(\beta )\right)$$

Use the distributive property to multiply -2 by \cos(\alpha )\cos(\beta )+\sin(\alpha )\sin(\beta ).

$$PQ^{2}=2-2\cos(\alpha )\cos(\beta )-2\sin(\alpha )\sin(\beta )$$

The equation is in standard form.

$$Q^{2}P=-2\sin(\alpha )\sin(\beta )-2\cos(\alpha )\cos(\beta )+2$$

Divide both sides by Q^{2}.

$$\frac{Q^{2}P}{Q^{2}}=\frac{2\left(-\cos(\alpha -\beta )+1\right)}{Q^{2}}$$

Dividing by Q^{2} undoes the multiplication by Q^{2}.

$$P=\frac{2\left(-\cos(\alpha -\beta )+1\right)}{Q^{2}}$$

Solve for P

$$\left\{\begin{matrix}P=-\frac{2\left(\sin(\alpha )\sin(\beta )+\cos(\alpha )\cos(\beta )-1\right)}{Q^{2}}\text{, }&Q\neq 0\\P\in \mathrm{R}\text{, }&\exists n_{1}\in \mathrm{Z}\text{ : }\alpha =2\pi n_{1}+\beta \text{ and }Q=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply -1 and 2 to get -2.

$$PQ^{2}=2-2\left(\cos(\alpha )\cos(\beta )+\sin(\alpha )\sin(\beta )\right)$$

Use the distributive property to multiply -2 by \cos(\alpha )\cos(\beta )+\sin(\alpha )\sin(\beta ).

$$PQ^{2}=2-2\cos(\alpha )\cos(\beta )-2\sin(\alpha )\sin(\beta )$$

The equation is in standard form.

$$Q^{2}P=-2\sin(\alpha )\sin(\beta )-2\cos(\alpha )\cos(\beta )+2$$

Divide both sides by Q^{2}.

$$\frac{Q^{2}P}{Q^{2}}=\frac{2\left(-\cos(\alpha -\beta )+1\right)}{Q^{2}}$$

Dividing by Q^{2} undoes the multiplication by Q^{2}.

$$P=\frac{2\left(-\cos(\alpha -\beta )+1\right)}{Q^{2}}$$

Solve for Q (complex solution)

$$\left\{\begin{matrix}Q=-iP^{-\frac{1}{2}}e^{\frac{i\beta -i\alpha }{2}}\left(e^{i\alpha -i\beta }-1\right)\text{; }Q=iP^{-\frac{1}{2}}e^{\frac{i\beta -i\alpha }{2}}\left(e^{i\alpha -i\beta }-1\right)\text{, }&P\neq 0\\Q\in \mathrm{C}\text{, }&\exists n_{1}\in \mathrm{Z}\text{ : }\alpha =2\pi n_{1}+\beta \text{ and }P=0\end{matrix}\right.$$