7. Given the function $y=6 \cos x$ defined on the interval $\left(0, \frac{\pi}{2}\right)$, its graph intersects with the graph of $y=5 \tan x$ at point $P$. A perpendicular line $P P_{1}$ is drawn from $P$ to the x-axis at point $P_{1}$. The line $P P_{1}$ intersects the graph of $y=\sin x$ at point $P_{2}$. Then the length of the line segment $P_{1} P_{2}$ is $\qquad$
$$
=7 . \frac{2}{3} \text {. }
$$

Let the x-coordinate of point $P$ be $x$. Then $6 \cos x=5 \tan x$.
Solving this, we get $\sin x=\frac{2}{3}$.
Given the condition, the length of $P_{1} P_{2}$ is $\frac{2}{3}$.