1. Given equation

We need to solve

Let

Since $x\in\left[-\frac12,\frac12\right]$, we have $2x\in[-1,1]$, so $\theta$ is well-defined.

Also,

Because $x\in[-1,1]$, we know $\sqrt{1-x^2}\in[0,1]$, hence

So the equation becomes

That is,

2. Range analysis

Now,

And since $\phi\in\left[0,\frac\pi2\right]$,

For

to hold with $\theta\in[0,\pi]$, the right-hand side must also lie in $[0,\pi]$. But $\pi+2\phi\ge \pi$, so the only possible case is

Thus,

This implies

Check in the original equation:

So even $x=0$ does not satisfy the equation.

Hence,

3. Required sum

We need

Since $S$ is empty, the sum over an empty set is

4. Compare with options

• Option A: $\pi-2\sin^{-1}\left(\frac{\sqrt3}{4}\right) \ne 0$
• Option B: $\pi-\sin^{-1}\left(\frac{\sqrt3}{4}\right) \ne 0$
• Option C: $-\frac{2\pi}{3} \ne 0$
• Option D: None

Therefore, the correct option is

5. Comparison with stored answer

Stored correct answer: D

Our derived answer: D

So they agree.