Find specific 4 curves touching $y=\cos10x+\cos21x$. The following is the graph of $y=\cos10x+\cos21x$.
You can see that there seems to be four curves that can touch this graph. I tried $y=\cos(x/2+\pi/2\pm\pi)+1$ and $y=-\cos(x/2\pm\pi/2)-1$:

But unfortunately, they cut the graph. What actually are that four curves touching the graph? Thanks.

Take the one that goes through $(0,2)$.  The peaks are roughly at $x=4n\pi/21$, call them $x_n=4n\pi/21+\beta_n$.  I will assume $\beta_n$ are small.
At these peaks, 
$$\frac{dy}{dx}=-10\sin10x-21\sin21x=0\\
10\sin\left(\frac{40n\pi}{21}+10\beta_n\right)+21\sin(4n\pi+21\beta_n)=0\\
10\left(\sin\frac{40n\pi}{21}+10\beta_n\cos\frac{40n\pi}{21}\right)+21^2\beta_n\approx0\\
\beta_n\approx\frac{10\sin(2n\pi/21)}{10^2\cos(2n\pi/21)+21^2}<\frac1{40}$$
The height of the peak is $\cos10x+\cos21x$, which is roughly
$$y=\cos\left(\frac{40n\pi}{21}+10\beta_n\right)+\cos(21\beta_n)\\
\approx\cos\left(\frac{2n\pi}{21}-10\beta_n\right)+1-(21\beta_n)^2/2\\
\approx\cos\left(\frac x2-\frac{21}2\beta_n\right)+1-5\beta_n\sin(x/2)\\
\approx\cos\left(\frac x2-\frac{11}2\beta_n\right)+1\\
\approx1+\cos\left(\frac x2-\frac18\sin\frac x2\right)\\
\approx1+\cos(\frac x2)+\frac18\sin^2\frac x2$$
