Solve the equation for \( x \), where \( x \) is in radians. State the exact solutions, if any, that lie in the interval \( [0,2 \pi) \). If the equation has no solutions, respond with DNE. \( \cos (4 x) \cos (x)=-\sin (4 x) \sin (x) \) in the interval \( [0,2 \pi) \). If the equation has no solutions, respond DNE. Separate muitiple solutions with a comma.

To solve the equation \( \cos(4x) \cos(x) = -\sin(4x) \sin(x) \), we can use the product-to-sum identities. The right-hand side can be rewritten as: \[ \cos(4x) \cos(x) + \sin(4x) \sin(x) = 0 \] This simplifies to: \[ \cos(4x + x) = 0 \quad \text{or} \quad \cos(5x) = 0 \] The solutions for \( \cos(5x) = 0 \) are given by: \[ 5x = \frac{\pi}{2} + n\pi \quad \text{for } n \in \mathbb{Z} \] From this, we can find \( x \): \[ x = \frac{\pi}{10} + \frac{n\pi}{5} \] Now we need to find the values of \( x \) in the interval \( [0, 2\pi) \). Setting \( n = 0, 1, 2, 3, 4, 5 \): - For \( n = 0 \): \( x = \frac{\pi}{10} \) - For \( n = 1 \): \( x = \frac{\pi}{10} + \frac{\pi}{5} = \frac{3\pi}{10} \) - For \( n = 2 \): \( x = \frac{\pi}{10} + \frac{2\pi}{5} = \frac{5\pi}{10} = \frac{\pi}{2} \) - For \( n = 3 \): \( x = \frac{\pi}{10} + \frac{3\pi}{5} = \frac{7\pi}{10} \) - For \( n = 4 \): \( x = \frac{\pi}{10} + \frac{4\pi}{5} = \frac{9\pi}{10} \) - For \( n = 5 \): \( x = \frac{\pi}{10} + \frac{5\pi}{5} = \frac{11\pi}{10} \) - For \( n = 6 \): \( x = \frac{\pi}{10} + \frac{6\pi}{5} = \frac{13\pi}{10} \) - For \( n = 7 \): \( x = \frac{\pi}{10} + \frac{7\pi}{5} = \frac{15\pi}{10} = \frac{3\pi}{2} \) - For \( n = 8 \): \( x = \frac{\pi}{10} + \frac{8\pi}{5} = \frac{17\pi}{10} \) Now, filtering the valid values of \( x \) within \( [0, 2\pi) \): The solutions are: \[ x = \frac{\pi}{10}, \frac{3\pi}{10}, \frac{\pi}{2}, \frac{7\pi}{10}, \frac{9\pi}{10}, \frac{11\pi}{10}, \frac{13\pi}{10}, \frac{3\pi}{2} \] So, the exact solutions in the interval \( [0, 2\pi) \) are: \(\frac{\pi}{10}, \frac{3\pi}{10}, \frac{\pi}{2}, \frac{7\pi}{10}, \frac{9\pi}{10}, \frac{11\pi}{10}, \frac{13\pi}{10}, \frac{3\pi}{2}\)