Question:

Step 1: Rewrite the equation in a more workable form. We start with the equation $\sin x \cdot \tan 4x = \cos x$. Since $\tan 4x = \frac{\sin 4x}{\cos 4x}$, we can write: $$\sin x \cdot \frac{\sin 4x}{\cos 4x} = \cos x$$ Step 2: Clear the denominator by multiplying both sides by $\cos 4x$. Assuming $\cos 4x \neq 0$ (which is necessary for $\tan 4x$ to be defined), we multiply both sides by $\cos 4x$: $$\sin x \sin 4x = \cos x \cos 4x$$ Step 3: Rearrange to apply the cosine addition formula. Rearranging the equation: $$\cos x \cos 4x - \sin x \sin 4x = 0$$ This matches the cosine addition formula: $\cos(A + B) = \cos A \cos B - \sin A \sin B$ Therefore: $$\cos(x + 4x) = 0$$ $$\cos 5x = 0$$ Step 4: Solve the trigonometric equation $\cos 5x = 0$. The general solution for $\cos \theta = 0$ is $\theta = \frac{\pi}{2} + k\pi$ where $k \in \mathbb{Z}$. Thus: $$5x = \frac{\pi}{2} + k\pi = \frac{(2k+1)\pi}{2}$$ $$x = \frac{(2k+1)\pi}{10}$$ Step 5: Find all solutions in the interval $(0, \pi)$. We need $0 < \frac{(2k+1)\pi}{10} < \pi$, which gives us $0 < 2k+1 < 10$, so $k \in \{0, 1, 2, 3, 4\}$. The candidate solutions are: - $k = 0$: $x = \frac{\pi}{10}$ - $k = 1$: $x = \frac{3\pi}{10}$ - $k = 2$: $x = \frac{5\pi}{10} = \frac{\pi}{2}$ - $k = 3$: $x = \frac{7\pi}{10}$ - $k = 4$: $x = \frac{9\pi}{10}$ Step 6: Verify that $\cos 4x \neq 0$ for each solution. We must check that none of these values make $\cos 4x = 0$ (since $\tan 4x$ must be defined). - At $x = \frac{\pi}{10}$: $4x = \frac{2\pi}{5}$, and $\cos\frac{2\pi}{5} \neq 0$ ✓ - At $x = \frac{3\pi}{10}$: $4x = \frac{6\pi}{5}$, and $\cos\frac{6\pi}{5} \neq 0$ ✓ - At $x = \frac{\pi}{2}$: $4x = 2\pi$, and $\cos 2\pi = 1 \neq 0$ ✓ - At $x = \frac{7\pi}{10}$: $4x = \frac{14\pi}{5}$, and $\cos\frac{14\pi}{5} \neq 0$ ✓ - At $x = \frac{9\pi}{10}$: $4x = \frac{18\pi}{5}$, and $\cos\frac{18\pi}{5} \neq 0$ ✓ All five values satisfy the constraint that $\cos 4x \neq 0$. **Final Answer:** The total number of solutions in $(0, \pi)$ is **5**. This corresponds to **Option 3: 5**.