= Solution
Substitution into the <circle> gives $x^2\sec^2x=25$, or $x=5|\cos x|$ since $x\geq0$. Necessarily $x\leq5$, and each admissible $x$ fixes a unique $y=x\tan x$.
On $[0,\pi/2)$, $x-5\cos x$ is strictly increasing from a negative value to a positive value, giving one root. On $(\pi/2,3\pi/2)$, the equation is $x+5\cos x=0$. This <function> is strictly convex, positive at the two interval ends and negative at $\pi$, giving exactly two roots.
There are none on the remaining possible interval $3\pi/2<x\leq5$. Put $\delta=x-3\pi/2>0$; then $\cos x=\sin\delta\leq\delta$, so
$$
x-5\cos x\geq\frac{3\pi}{2}-4\delta
\geq\frac{15\pi}{2}-20>0.
$$
Hence the number of simultaneous real solutions is \b[$3$], as marked in the preceding graph.
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