Solve Tcosθ=4mg

$$T \cos \theta = 4 m g$$

Solve for T

$$\left\{\begin{matrix}T=\frac{4gm}{\cos(\theta )}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}+\frac{\pi }{2}\\T\in \mathrm{R}\text{, }&\left(m=0\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}+\frac{\pi }{2}\right)\text{ or }\left(g=0\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}+\frac{\pi }{2}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

The equation is in standard form.

$$\cos(\theta )T=4gm$$

Divide both sides by \cos(\theta ).

$$\frac{\cos(\theta )T}{\cos(\theta )}=\frac{4gm}{\cos(\theta )}$$

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

$$T=\frac{4gm}{\cos(\theta )}$$

Solve for g

$$\left\{\begin{matrix}g=\frac{T\cos(\theta )}{4m}\text{, }&m\neq 0\\g\in \mathrm{R}\text{, }&\left(T=0\text{ or }\exists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}+\frac{\pi }{2}\right)\text{ and }m=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$4mg=T\cos(\theta )$$

Divide both sides by 4m.

$$\frac{4mg}{4m}=\frac{T\cos(\theta )}{4m}$$

Dividing by 4m undoes the multiplication by 4m.

$$g=\frac{T\cos(\theta )}{4m}$$