This part answered :-
How does one determine what is $t$ when we have two trigonometric functions added to each other? Like $\tan(x) + \sin(2x)$ \begin{align} \tan(x) &= \tan(x + \pi) \\ \sin(2x) &= \sin(2x + \pi) \end{align}
Do we take the $\pi$ as $t$ because they have it in common?
Answered section end
Is the sum of trigonometric periodic function often periodic or always and why?(i would like to share my take on this one from what i read and Deduced)
And how do we determine that this function is periodic and find $t$? $$ \tan(x) \sin(3x) $$ since \begin{align} \tan(x) &= \tan(x + \pi) \\ \sin(3x) & = \sin(2x + 2\pi/3) \end{align} And thx in advance
I read the resources above and tried to understand the two participants shares , I understand now how to determine t for the sum of two functions , As said above we try to find the least common multiple of their period . for example:-
$\sin(2x) + \cos(3x)$ \begin{align} \cos(3x) &= \cos(3x +2\pi/3)&=\cos(3x +tn)\\ \sin(2x) &= \sin(2x + \pi) &= \sin(2x +tm)\\ \end{align} $m>0 ,n>0$ and they are whole numbers
So in this case we have the least common multiple when n is 3 and m is 2 Resulting in t equals $2\pi $ for the sum of the two functions .
I will try to share what I have deduced in regards to if the sum of two Periodic functions is always periodic since I now have a slightly better understanding of the topic ,
The answer is no, there does not exist non zero integers $a$,$b$ such that $a t_1=bt_2$
For example cos($\pi x$)+sin($2x$). We can see that $a \times 2=b \times \pi$ does not exist, Hence the answer is a non periodic function
If what I deduced is incorrect any correction is appreciated,
I am left with a couple of questions unanswered, I will try to edit the original version to highlight them.