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

Question

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.

Answer 1

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 $$\frac{2\pi}{4} = \frac{\pi}{2}$$

Answer 2

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

Answer 3

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