Step 1: Clear the fraction.
Write $\tan 5x = \sin 5x/\cos 5x$ and get $\cos 3x \sin 5x - \sin 7x \cos 5x = 0$, with $\cos 5x \neq 0$.
Step 2: Use sum and difference formulas.
$2\cos 3x \sin 5x = \sin 8x + \sin 2x$ and $2\sin 7x \cos 5x = \sin 12x + \sin 2x$. Subtracting gives $\sin 8x - \sin 12x = 0$.
Step 3: Factorise.
\[ 2\cos 10x \sin(-2x) = 0 \Rightarrow \cos 10x \sin 2x = 0 \]
Step 4: Solve each factor.
$\sin 2x = 0$ gives $x = 0$ in the interval. $\cos 10x = 0$ gives $10x = \pi/2, 3\pi/2, \ldots$ so $x = \pi/20, 3\pi/20, 5\pi/20, 7\pi/20, 9\pi/20$, all below $\pi/2$.
Step 5: Check the exclusions.
At these points $5x = \pi/4, 3\pi/4, 5\pi/4, 7\pi/4, 9\pi/4$, so $\cos 5x \neq 0$. All are valid.
Final Answer:
We get 1 + 5 = 6 solutions, option (D).
\[ \boxed{6} \]