MHB Periodic Function: Prove Smallest Positive Period

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The function f(x) = tg(11x/34) + ctg(13x/54) has a period of 918π, which is the least common multiple of the individual periods of its components. The tangent function repeats every 34π/11 and the cotangent every 54π/13. To prove that 918π is the smallest positive period, one must examine the points where the function becomes undefined, which occur at specific intervals that do not repeat before 918π. This analysis shows that no smaller period can satisfy the conditions of the function. Thus, 918π is confirmed as the smallest positive period of f.
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One period of the function $$f(x)=\operatorname{tg}\frac{11x}{34}+\operatorname{ctg}\frac{13x}{54}$$ is $$918\pi.$$ Please help me to prove that this is the smallest positive period. I can not use the most of trigonometric identities.
 
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Andrei said:
One period of the function $$f(x)=\operatorname{tg}\frac{11x}{34}+\operatorname{ctg}\frac{13x}{54}$$ is $$918\pi.$$ Please help me to prove that this is the smallest positive period. I can not use the most of trigonometric identities.
The tangent and cotangent functions both have period $\pi$. So the function $\tan\frac{11x}{34}$ will repeat at intervals $\frac{34\pi}{11}$, and $\cot\frac{13x}{54}$ will repeat at intervals $\frac{54\pi}{13}$. You need to find the least common multiple of those two intervals.
 
Opalg said:
You need to find the least common multiple of those two intervals.
$$T=918\pi$$ is the least common multiple of those periods. I found this. But why it is the smallest positive period of $$f$$?
For example, I consider the functions $$f_1(x)=\sin x$$ and $$f_2(x)=\operatorname{tg} x-\sin x$$, which both have $$2\pi$$ as main period. But then $$\pi$$ is the main period of $$f_1+f_2$$, which is not the least common multiple of $$2\pi.$$
 
Andrei said:
$$T=918\pi$$ is the least common multiple of those periods. I found this. But why it is the smallest positive period of $$f$$?
For example, I consider the functions $$f_1(x)=\sin x$$ and $$f_2(x)=\operatorname{tg} x-\sin x$$, which both have $$2\pi$$ as main period. But then $$\pi$$ is the main period of $$f_1+f_2$$, which is not the least common multiple of $$2\pi.$$
You could look at the set of points at which $f(x)$ becomes infinite (or is undefined). That will happen at the points $x = \bigl(\frac{34}{11}k + \frac12\bigr)\pi$ and $x = \frac{54}{13}k\pi$ (and nowhere else). I think you should find that that set of points does not repeat at intervals of less than $918\pi$.
 
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