Question

Find the periods $\cos \left(\cos x\right)+\cos \left(\sin x\right)$

• A. $\pi /2$
• B. $\pi$
• C. $2\pi$
• D. $4\pi$

Answer 1

The correct option is A $\pi /2$

$\cos \left(\cos (\pi +x)\right)=\cos (-\cos x)=\cos \left(\cos x\right)$ $\cos \left(\sin (\pi +x)\right)=\cos (-\sin x)=\cos \left(\sin x\right)$ Thus both the functions f and g are periodic functions of period $\pi$ and hence period of $f+g$ is L.C.M of $\pi and\pi i.e.,\pi .$ But look there exists a number$\frac{\pi }{2}$ less than $\pi$ such that $\cos \left[\cos (\frac{\pi }{2}+x)\right]+\cos \left[\sin (\frac{\pi }{2}+x)\right]$ $=\cos (-\sin x)+\cos \left(\cos x\right)$ $=\cos \left(\sin x\right)+\cos \left(\cos x\right)$ $=\cos \left(\cos x\right)+\cos \left(\sin x\right)$ Above shows tha period is $\frac{\pi }{2}$ but not $\pi$ as found above.