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Prove that the period of the tan and cot function is $\pi$.

I want to solve this question by using basic periodic function definition and trig point function... Here is my try :

$\cos \theta = \cos (\theta + 2\pi n)$ where $n\in Z$. and then by method of contradiction, I have proved $\pi$ can't be period of the cosine function. Hence $2\pi$ is the principle period of cosine function.

Similarly, I have shown $2\pi$ is the principal period of the sine function.

In my textbook, it is written “For the time being we shall assume that the values of tan and cot functions do not change if $\theta$ increases or decreases by an integral multiple of $\pi$. and the principal period of the tan and cot function is $\pi$.” [But what is the proof of this? ]

It is obvious that $\pi$ is a period of tan and cot functions but how can I show $\pi$ is the principal period? I have no idea.

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If there were a smaller period, then there would be a number $\alpha$ satisfying $0 < \alpha < \pi$ for which $\tan(\alpha) = \tan(0) = 0$. But this implies that $\sin(\alpha) = 0$ (since $\tan = \sin/\cos$), which is a contradiction.

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HINT:

If $\tan A=\tan B,$

$$\frac{\sin A}{\cos A}=\frac{\sin B}{\cos B}\implies \sin A\cos B-\cos A\sin B=0\implies \sin(A-B)=0$$

$$\implies A-B=n\pi$$ where $n$ is any integer

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