Solve operatorname{cotg}(2x-pi)=operatorname{cotg}(x+pi/3)

$$\operatorname { cotg } ( 2 x - \pi ) = \operatorname { cotg } ( x + \frac { \pi } { 3 } )$$

Solve for x

$$\left\{\begin{matrix}x=\frac{4\pi }{3}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }g=\pi n_{1}\\x\in \mathrm{R}\text{, }&\exists n_{2}\in \mathrm{Z}\text{ : }g=\pi n_{2}+\frac{\pi }{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 3.

$$3\cot(g)\left(2x-\pi \right)=3\cot(g)\left(x+\frac{\pi }{3}\right)$$

Use the distributive property to multiply 3\cot(g) by 2x-\pi .

$$6\cot(g)x-3\cot(g)\pi =3\cot(g)\left(x+\frac{\pi }{3}\right)$$

Use the distributive property to multiply 3\cot(g) by x+\frac{\pi }{3}.

$$6\cot(g)x-3\cot(g)\pi =3\cot(g)x+3\cot(g)\times \frac{\pi }{3}$$

Express 3\times \frac{\pi }{3} as a single fraction.

$$6\cot(g)x-3\cot(g)\pi =3\cot(g)x+\frac{3\pi }{3}\cot(g)$$

Cancel out 3 and 3.

$$6\cot(g)x-3\cot(g)\pi =3\cot(g)x+\pi \cot(g)$$

Subtract 3\cot(g)x from both sides.

$$6\cot(g)x-3\cot(g)\pi -3\cot(g)x=\pi \cot(g)$$

Combine 6\cot(g)x and -3\cot(g)x to get 3\cot(g)x.

$$3\cot(g)x-3\cot(g)\pi =\pi \cot(g)$$

Add 3\cot(g)\pi to both sides.

$$3\cot(g)x=\pi \cot(g)+3\cot(g)\pi$$

Combine \pi \cot(g) and 3\cot(g)\pi to get 4\pi \cot(g).

$$3\cot(g)x=4\pi \cot(g)$$

Divide both sides by 3\cot(g).

$$\frac{3\cot(g)x}{3\cot(g)}=\frac{4\pi \cot(g)}{3\cot(g)}$$

Dividing by 3\cot(g) undoes the multiplication by 3\cot(g).

$$x=\frac{4\pi \cot(g)}{3\cot(g)}$$

Divide 4\pi \cot(g) by 3\cot(g).

$$x=\frac{4\pi }{3}$$

Solve for g

$$\left\{\begin{matrix}\\g=\pi n_{1}+\frac{\pi }{2}\text{, }n_{1}\in \mathrm{Z}\text{, }&\text{unconditionally}\\g\neq \pi n_{2}\text{, }\forall n_{2}\in \mathrm{Z}\text{, }&x=\frac{4\pi }{3}\end{matrix}\right.$$