Step 1 :Let \(\theta = \cos^{-1}(\frac{w}{6})\)
Step 2 :We know that \(\cos(2\theta) = 1 - 2\sin^2(\theta)\)
Step 3 :We also know that \(\sin^2(\theta) = 1 - \cos^2(\theta)\)
Step 4 :So, we can substitute \(\cos^2(\theta)\) with \((\frac{w}{6})^2\)
Step 5 :Substituting these values into the equation for \(\cos(2\theta)\), we get \(\cos(2\theta) = 1 - 2(1 - (\frac{w}{6})^2)\)
Step 6 :Simplifying this, we get \(\cos(2\theta) = 2(\frac{w}{6})^2 - 1\)
Step 7 :So, \(\cos \left(2 \cos ^{-1} \frac{w}{6}\right)=\boxed{2\left(\frac{w}{6}\right)^2 - 1}\)