Solve (1-tanx)^2=1-2tanx+tan^2x

$$( 1 - \tan x ) ^ { 2 } = 1 - 2 \tan x + \tan ^ { 2 } x$$

Solve for x (complex solution)

$$x\in \mathrm{C}$$

$$\nexists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}+\frac{\pi }{2}$$

Solution Steps

Subtract 1-2\tan(x) from both sides.

$$\left(1-\tan(x)\right)^{2}-\left(1-2\tan(x)\right)=\left(\tan(x)\right)^{2}$$

Subtract \left(\tan(x)\right)^{2} from both sides.

$$\left(1-\tan(x)\right)^{2}-\left(1-2\tan(x)\right)-\left(\tan(x)\right)^{2}=0$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-\tan(x)\right)^{2}.

$$1-2\tan(x)+\left(\tan(x)\right)^{2}-\left(1-2\tan(x)\right)-\left(\tan(x)\right)^{2}=0$$

To find the opposite of 1-2\tan(x), find the opposite of each term.

$$1-2\tan(x)+\left(\tan(x)\right)^{2}-1+2\tan(x)-\left(\tan(x)\right)^{2}=0$$

Subtract 1 from 1 to get 0.

$$-2\tan(x)+\left(\tan(x)\right)^{2}+2\tan(x)-\left(\tan(x)\right)^{2}=0$$

Combine -2\tan(x) and 2\tan(x) to get 0.

$$\left(\tan(x)\right)^{2}-\left(\tan(x)\right)^{2}=0$$

Combine \left(\tan(x)\right)^{2} and -\left(\tan(x)\right)^{2} to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{C}$$

Solve for x

$$x\neq \pi n_{1}+\frac{\pi }{2}$$

$$\forall n_{1}\in \mathrm{Z}$$

Solution Steps

Subtract 1-2\tan(x) from both sides.

$$\left(1-\tan(x)\right)^{2}-\left(1-2\tan(x)\right)=\left(\tan(x)\right)^{2}$$

Subtract \left(\tan(x)\right)^{2} from both sides.

$$\left(1-\tan(x)\right)^{2}-\left(1-2\tan(x)\right)-\left(\tan(x)\right)^{2}=0$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-\tan(x)\right)^{2}.

$$1-2\tan(x)+\left(\tan(x)\right)^{2}-\left(1-2\tan(x)\right)-\left(\tan(x)\right)^{2}=0$$

To find the opposite of 1-2\tan(x), find the opposite of each term.

$$1-2\tan(x)+\left(\tan(x)\right)^{2}-1+2\tan(x)-\left(\tan(x)\right)^{2}=0$$

Subtract 1 from 1 to get 0.

$$-2\tan(x)+\left(\tan(x)\right)^{2}+2\tan(x)-\left(\tan(x)\right)^{2}=0$$

Combine -2\tan(x) and 2\tan(x) to get 0.

$$\left(\tan(x)\right)^{2}-\left(\tan(x)\right)^{2}=0$$

Combine \left(\tan(x)\right)^{2} and -\left(\tan(x)\right)^{2} to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{R}$$