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I have always been taught that in the scenario of a Sine,Tan,Cos function

of $f(x) = a\sin b(x+c) +d$, the period of the sine and cos functions $= \dfrac{2\pi}{b}$, and the period for the tan function $= \dfrac{\pi}{b}$

I don't see how this would apply to trigonometric functions that have powers or trig functions multiplied within the function

e.g $\sin x\tan x + \cos x$...what would be the period?

$\cos^2x\tan x - \sin x$...what would be the period?

Thanks

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If one period is a rational multiple of the other then there will be a common period that is the lowest common multiple of the two periods, however if one period is a rational number and the other is irrational then the result is not even periodic as they cannot cycle through integer multiples of their respective periods in any given interval –  Graham Hesketh Aug 20 '13 at 23:46

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Any expression comprised of functions that all have the same period $t$ with respect to a certain variable is either constant or has a period which divides $t$. So for example, $\sin(x)^2$ has period $\pi$ which divides $2 \cdot \pi$, the period of $\sin(x)$.

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