Rewrite $\cos \left(2 \tan ^{-1} 6 u\right)$ as an algebraic expression in $u$.

Step 1 ：Given the expression is \(\cos \left(2 \tan ^{-1} 6 u\right)\).

Step 2 ：We can use the identity \(\cos(2x) = 1 - 2\sin^2(x)\) and the fact that \(\tan(x) = \sin(x)/\cos(x)\) to rewrite the expression in terms of u.

Step 3 ：Substitute \(x = \tan^{-1}(6u)\) into the identity, we get \(\cos(2x) = 1 - 2\sin^2(x) = 1 - 2\left(\frac{6u}{\sqrt{1+(6u)^2}}\right)^2\).

Step 4 ：Simplify the expression, we get \(-72u^2/(36u^2 + 1)^2 + 1\).

Step 5 ：Further simplify the expression, we get \((1296u^4 + 1)/(1296u^4 + 72u^2 + 1)\).

Step 6 ：So, the expression \(\cos \left(2 \tan ^{-1} 6 u\right)\) can be rewritten as \(\boxed{\frac{1296u^4 + 1}{1296u^4 + 72u^2 + 1}}\) in terms of u.