# What is the General Solution of $$tan \theta$$ = $$tan \alpha$$ ?

## Solution :

The general solution of $$tan \theta$$ = $$tan \alpha$$ is given by $$\theta$$ = $$n\pi + \alpha$$,  n $$\in$$ Z.

Proof :

We have, $$tan \theta$$ = $$tan \alpha$$

$$\implies$$  $$sin \theta\over cos \theta$$ = $$sin \alpha\over cos \alpha$$

$$\implies$$  $$sin \theta cos \alpha$$ – $$cos \theta sin \alpha$$ = 0

$$\implies$$  $$sin (\theta – \alpha)$$ = 0

$$\implies$$   $$\theta – \alpha$$ = $$n\pi$$, n $$\in$$ Z

$$\implies$$    $$\theta$$ = $$n\pi + \alpha$$, n $$\in$$ Z

Remark : Since $$tan \theta$$ = $$tan \alpha$$ is equivalent to $$cot \theta$$ = $$cot \alpha$$. So, general solutions of $$cot \theta$$ = $$cot \alpha$$ and $$tan \theta$$ = $$tan \alpha$$ are same.