${\cos }^{-1}[\cos \left(\frac{6\pi }{7}\right)]$

Step 1 ：The problem is asking for the inverse cosine of the cosine of $\frac{6\pi }{7}$. The inverse cosine function, or arccosine, is the function that takes a number in the range of -1 to 1 (the range of the cosine function) and returns the angle in radians whose cosine is that number. The range of the arccosine function is $[0,\pi ]$.

Step 2 ：The cosine function is periodic with period $2\pi$, so $\cos (\frac{6\pi }{7})$ is the same as $\cos (\frac{6\pi }{7}-2\pi )$, $\cos (\frac{6\pi }{7}-4\pi )$, and so on.

Step 3 ：However, since the range of the arccosine function is $[0,\pi ]$, we need to find a value equivalent to $\frac{6\pi }{7}$ that falls within this range. We can do this by subtracting multiples of $2\pi$ from $\frac{6\pi }{7}$ until we get a value within the range of the arccosine function.

Step 4 ：The value of $\frac{6\pi }{7}$ after subtracting multiples of $2\pi$ until it falls within the range $[0,\pi ]$ is approximately 2.69. This is the value that the arccosine function will return when given $\cos (\frac{6\pi }{7})$ as input.

Step 5 ：Final Answer: ${\cos }^{-1}[\cos \left(\frac{6\pi }{7}\right)]=\boxed{{\displaystyle 2.69}}$ (approximately)