Mathematics - Functions Question with Solution | TestHub

For a function $f(x)$ to be periodic with period $T$, $f(x+T)=f(x)$ for all $x$. First, check for $2\pi$: $f(x+2\pi )=\cos (\sin (x+2\pi ))=\cos (\sin x)=f(x)$. So $2\pi$ is a period. Next, check for $\pi$: $f(x+\pi )=\cos (\sin (x+\pi ))=\cos (-\sin x)$. Since $\cos (-\theta )=\cos (\theta )$, we have $f(x+\pi )=\cos (\sin x)=f(x)$. So $\pi$ is also a period. Finally, check for $\pi /2$: $f(x+\pi /2)=\cos (\sin (x+\pi /2))=\cos (\cos x)$. This is not equal to $\cos (\sin x)$ for all $x$. For example, if $x=0$, $f(0)=\cos (\sin 0)=\cos (0)=1$. But $f(0+\pi /2)=\cos (\cos 0)=\cos (1)$, which is not $1$. Therefore, $\pi /2$ is not a period. The fundamental period is the smallest positive value of $T$, which is $\pi$.