# What is the period of $f(x) = \sin^4(x)+ \cos^4(x)$?

This is an elementary problem but I'm just not getting the right answer.

My reasoning is as follows: The period of $g(x) = \sin^4(x)$ is $\pi$ and that of $h(x) = \cos^4(x)$ is $π$ as well, so the period of the function $f(x) = \sin^4(x) + \cos^4(x)$ should be the LCM, which would be $\pi$. Plotting the function, however, shows that the period is $\frac{\pi}{2}$.

Why is that? What am I missing here? I'm sorry if this is trivial, but I'm not able to figure it out.

• Hint: $1=(\sin(x)^2+\cos(x)^2)^2=\sin(x)^4+\cos(x)^4+\sin(2x)^2/2=\sin(x)^4+\cos(x)^4+(1-\cos(4x))/4$ – Raymond Manzoni Nov 12 '15 at 16:37
• Generally speaking, the sum of two functions of period $a$ is not necessarily of period $a$. Just take $f-f=0$, for instance. Or less trivially, the difference of two trigonometric polynomials $f,g$ which have the same first few terms. – Jean-Claude Arbaut Nov 12 '15 at 16:41
• @RaymondManzoni That is a "moster" of a "Hint." But +1 for the comment. – Mark Viola Nov 12 '15 at 16:45
• Thanks @Dr.MV (for the "moster" as well :-)) – Raymond Manzoni Nov 12 '15 at 16:47
• @RaymondManzoni My pleasure. ;-)) – Mark Viola Nov 12 '15 at 16:52

After some nice trig manipulations, you can find that

$$\sin^4(x)+\cos^4(x) = \frac{\cos(4x)+3}{4}$$

which has a period of $$\frac{2\pi}{4} = \frac{\pi}{2}$$

• Solid answer! +1 – Mark Viola Nov 12 '15 at 16:43
• It's "solid" for those of us who already know how to do it. – The Chaz 2.0 Nov 12 '15 at 16:48
• Nice edit :) +1 – The Chaz 2.0 Nov 12 '15 at 16:52

$$\sin^4x+\cos^4x=(\sin^2x+\cos^2x)^2-2\sin^2x\cos^2x=1-\frac12\sin^22x$$ as a result has a period of: $$\frac{2\pi}{4}=\frac{\pi}{2}$$

If $f$ and $g$ have period $\pi$, then $f + g$ has period of at most $\pi$. For example, $f = \sin$ and $g = -f$.

• Thank you! So f(x) can have a period of at most the LCM of the periods of g(x) and h(x) here? – Utkarsh Pant Nov 12 '15 at 16:40
• @UtkarshPant: well, not sure LCM is defined well for non-integers, but if you say that $z = \mathrm{LCM}(x,y)$ is smallest positive number such that $z/x, z/y\in \Bbb Z$ then yes. That just follows from the definition of periodicity. – Ilya Nov 12 '15 at 16:43
• I don't see how your example is related to your first sentence. A constant function has any nonzero number as a period, thus it's certainly not at most $\pi$. – Jean-Claude Arbaut Nov 12 '15 at 18:43