Step 1: Rewrite using sin 3x
Write $\cos^2\theta=1-\sin^2\theta$. The equation becomes $\sin3\theta-\tfrac34+\sin^2\theta=0$.
Step 2: Count the crossings
Plot $g(\theta)=\sin3\theta+\sin^2\theta-\tfrac34$. It is negative at $0$, touches near zero around $\pi/6$ and crosses twice in $(0,\pi/3)$. It is negative on $(\pi/3,2\pi/3)$ and crosses twice again in $(2\pi/3,\pi)$.
A numerical scan confirms exactly 4 sign changes, option (C).
Final Answer:
Four solutions exist in $[0,\pi]$, option (C).
\[ \boxed{4} \]