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

## Solution :

The general solution of $$cos \theta$$ = $$cos \alpha$$ is given by $$\theta$$ = $$2n\pi \pm \alpha$$,  n $$\in$$ Z.

Proof :

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

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

$$\implies$$   -$$2 sin ({\theta + \alpha\over 2}) sin({\theta – \alpha\over 2})$$ = 0

$$\implies$$  $$sin ({\theta + \alpha\over 2})$$ = 0   or,   $$cos({\theta – \alpha\over 2})$$ = 0

$$\implies$$  $${\theta + \alpha\over 2}$$ = $$n\pi$$   or  $${\theta – \alpha\over 2}$$ = $$n\pi$$ ,  n $$\in$$ Z

$$\implies$$   $$\theta$$  =  $$2n\pi – \alpha$$, $$\in$$  Z   or,  $$\theta$$  =  $$2n\pi + \alpha$$,  n $$\in$$ Z.

$$\implies$$    $$\theta$$ = $$2n\pi \pm \alpha$$,  where n $$\in$$ Z.

Remark : The equation $$sec \theta$$ = $$sec \alpha$$ is equivalent to $$cos \theta$$ = $$cos \alpha$$. Thus, $$sec \theta$$ = $$sec \alpha$$ and $$cos \theta$$ = $$cos \alpha$$ have the same general solution.