The number of real solution of $$\cos(\cos (\cos x))=\sin(\sin(\sin x))$$
Try: Let $f(x)=\sin(\sin x))$ and $g(x)=\cos(\cos x)$
Then $$f(x)-g(x)=\sin(\sin x)-\cos(\cos x)$$
$$=\sin(\sin x)-\sin\bigg(\frac{\pi}{2}-\cos x\bigg)$$
$$=2\cos\bigg(\frac{\sin x-\cos x+90^\circ}{2}\bigg)\sin\bigg(\frac{\sin x+\cos x-90^\circ}{2} \bigg)$$
Could some help me to solve it, Thanks
Best Answer
They are infinite. Note that $$f(x)=\cos(\cos (\cos x))-\sin(\sin(\sin x))$$ is a continuous $2\pi$-periodic function. Moreover $$f(0)=f(\pi)=\cos(\cos(1))>0\quad\text{and}\quad f(\pi/2)=\cos(1)-\sin(\sin(1))<0,$$ so, by continuity, in each interval $[2\pi k,2\pi(k+1))$ with $k\in\mathbb{Z}$, there are at least two solutions: $x_k\in (2\pi k,2\pi k +\pi/2)$ and $y_k\in (2\pi k +\pi/2, 2\pi k+\pi)$.