Solve y=xcosθ+ysinθ)sinθ+(xcosθ+ysinθ)^2cosθ

$$y = x \cos \theta + y \sin \theta ) \sin \theta + ( x \cos \theta + y \sin \theta ) ^ { 2 } \cos \theta$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{y\left(\sin(\theta )-1\right)}{\cos(\theta )}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}+\frac{\pi }{2}\\x\in \mathrm{R}\text{, }&\exists n_{2}\in \mathrm{Z}\text{ : }\theta =2\pi n_{2}+\frac{\pi }{2}\text{ or }\left(y=0\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}+\frac{\pi }{2}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$x\cos(\theta )+y\sin(\theta )=y$$

Subtract y\sin(\theta ) from both sides.

$$x\cos(\theta )=y-y\sin(\theta )$$

The equation is in standard form.

$$\cos(\theta )x=-y\sin(\theta )+y$$

Divide both sides by \cos(\theta ).

$$\frac{\cos(\theta )x}{\cos(\theta )}=\frac{y\left(-\sin(\theta )+1\right)}{\cos(\theta )}$$

Dividing by \cos(\theta ) undoes the multiplication by \cos(\theta ).

$$x=\frac{y\left(-\sin(\theta )+1\right)}{\cos(\theta )}$$

Solve for y

$$\left\{\begin{matrix}y=\frac{x\cos(\theta )}{-\sin(\theta )+1}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =2\pi n_{1}+\frac{\pi }{2}\\y\in \mathrm{R}\text{, }&\exists n_{1}\in \mathrm{Z}\text{ : }\theta =2\pi n_{1}+\frac{\pi }{2}\end{matrix}\right.$$