If $\sin \left(P\right)=\frac{1}{6}$ and $\cos \left(P\right)=\frac{\sqrt{35}}{6}$, find $\cot \left(P\right)$ using trigonometric identities.

Is the following trigonometric equation an identity?

$\cot \left(2x\right)=\frac{{\cot }^{2}x-1}{2\cot x}$