Solve xcosbeta=1/2(sin(alpha-beta)+sin(alpha+beta))

$$x \cos \beta = \frac { 1 } { 2 } ( \sin ( \alpha - \beta ) + \sin ( \alpha + \beta ) )$$

Solve for x

$$\left\{\begin{matrix}\\x=\sin(\alpha )\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&\exists n_{1}\in \mathrm{Z}\text{ : }\beta =\pi n_{1}+\frac{\pi }{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply \frac{1}{2} by \sin(\alpha -\beta )+\sin(\alpha +\beta ).

$$x\cos(\beta )=\frac{1}{2}\sin(\alpha -\beta )+\frac{1}{2}\sin(\alpha +\beta )$$

The equation is in standard form.

$$\cos(\beta )x=\frac{\sin(\alpha +\beta )+\sin(\alpha -\beta )}{2}$$

Divide both sides by \cos(\beta ).

$$\frac{\cos(\beta )x}{\cos(\beta )}=\frac{\sin(\alpha )\cos(\beta )}{\cos(\beta )}$$

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

$$x=\frac{\sin(\alpha )\cos(\beta )}{\cos(\beta )}$$

Divide \sin(\alpha )\cos(\beta ) by \cos(\beta ).

$$x=\sin(\alpha )$$

Solve for α

$$\left\{\begin{matrix}\alpha =-\arcsin(x)+2\pi n_{2}+\pi \text{, }n_{2}\in \mathrm{Z}\text{; }\alpha =\arcsin(x)+2\pi n_{3}\text{, }n_{3}\in \mathrm{Z}\text{, }&|x|\leq 1\\\alpha \in \mathrm{R}\text{, }&\exists n_{1}\in \mathrm{Z}\text{ : }\beta =\pi n_{1}+\frac{\pi }{2}\end{matrix}\right.$$