The curves $C_1: y_1=1-\cos x, x \in(0, \pi)$ and $C_2: y=\frac{\sqrt{3}}{2}|x|+a$ will touch each-other if

$a=\frac{3}{2}-\frac{\pi}{\sqrt{3}}$

Both the curve touch each other at $\sin x=\frac{\sqrt{3}}{2} \Rightarrow x=\frac{\pi}{3}$ or $\frac{2 \pi}{3}$

Hence point of contact is $\left(\frac{\pi}{3}, \frac{1}{2}\right)$ or $\left(\frac{2 \pi}{3}, \frac{3}{2}\right)$

For $\left(\frac{\pi}{3}, \frac{1}{3}\right)$, we get $a=\frac{1}{2}-\frac{\pi}{2 \sqrt{3}}$

For $\left(\frac{2 \pi}{3}, \frac{3}{2}\right)$, we get $a=\frac{3}{2}-\frac{\pi}{\sqrt{3}}$