Solve 0=N_{2}-mgcos(alpha)

$$0 = N _ { 2 } - m g \cos ( \alpha )$$

Solve for g

$$\left\{\begin{matrix}g=\frac{N_{2}}{m\cos(\alpha )}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }\alpha =\pi n_{1}+\frac{\pi }{2}\text{ and }m\neq 0\\g\in \mathrm{R}\text{, }&\left(m=0\text{ or }\exists n_{1}\in \mathrm{Z}\text{ : }\alpha =\pi n_{1}+\frac{\pi }{2}\right)\text{ and }N_{2}=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$N_{2}-mg\cos(\alpha )=0$$

Subtract N_{2} from both sides. Anything subtracted from zero gives its negation.

$$-mg\cos(\alpha )=-N_{2}$$

The equation is in standard form.

$$\left(-m\cos(\alpha )\right)g=-N_{2}$$

Divide both sides by -m\cos(\alpha ).

$$\frac{\left(-m\cos(\alpha )\right)g}{-m\cos(\alpha )}=-\frac{N_{2}}{-m\cos(\alpha )}$$

Dividing by -m\cos(\alpha ) undoes the multiplication by -m\cos(\alpha ).

$$g=-\frac{N_{2}}{-m\cos(\alpha )}$$

Divide -N_{2} by -m\cos(\alpha ).

$$g=\frac{N_{2}}{m\cos(\alpha )}$$

Solve for N_2

$$N_{2}=gm\cos(\alpha )$$