A geometric sequence $(a_n)$ has $a_1=\sin x$, $a_2=1$, and $a_3=\tan x$ for some real number $x$. What is $a_6$?

1. In a geometric sequence $a_2^2=a_1a_3$, so $1=\sin x\tan x=\dfrac{\sin^2x}{\cos x}$. Hence $\sin^2x=\cos x$.
2. With $c=\cos x$: $1-c^2=c$, so $c^2+c-1=0$ and $c=\frac{\sqrt5-1}2$ (positive since $\cos x=\sin^2x>0$).
3. The common ratio is $\dfrac1{\sin x}$, so $a_n=\sin x\cdot\left(\dfrac1{\sin x}\right)^{n-1}=\sin^{2-n}x$ and $a_6=\dfrac1{\sin^4x}=\dfrac1{c^2}$.
4. From $c^2+c=1$ we get $\frac1c=c+1=\frac{1+\sqrt5}2$, so $a_6=\left(\dfrac{1+\sqrt5}2\right)^2=\boxed{\dfrac{3+\sqrt5}2}$.