The Unit Circle and Circular Functions · Superchallenge problem

Let $O=(0,0)$ and $A=(1,0)$. For $0<t<\dfrac{\pi}{2}$, let $P$ be the terminal point of $t$ on the unit circle, $Q$ the foot of the perpendicular from $P$ to the $x$-axis, and $T$ the point where ray $OP$ meets the tangent line $x=1$. If the area of triangle $OAT$ is $8$ times the area of triangle $OQP$, find $\cos t$.