Drag and drop the correct answer into the space(s) provided, Determine all solutions of the equation \( 8 \sin x+7=10 \sin x+1 \), \[ x=\square, n \in \mathbb{Z} \] \( \frac{\pi}{2}+2 n \pi \) \( \frac{\pi}{2}+n \pi \) \( \pi+2 n \pi \) \( \pi+n \pi \)

To solve the equation \( 8 \sin x + 7 = 10 \sin x + 1 \), we first rearrange it: \[ 8 \sin x + 7 - 1 = 10 \sin x \implies 8 \sin x + 6 = 10 \sin x \] Now, simplify: \[ 6 = 2 \sin x \implies \sin x = 3 \] Since the sine function has a range of \([-1, 1]\), the equation \(\sin x = 3\) has no solution. Thus, we conclude that there are no values of \(x\) that satisfy this equation. So when prompted for solutions of the form \( x=\square, n \in \mathbb{Z} \), the correct response is that there are no valid answers from the options provided. Now, let's fill our box: \( x=\emptyset, n \in \mathbb{Z} \)