\( \begin{array}{ll}\text { (1) } \cos 5 x-\cos 15 x=2 \sin -10 x \sin 5 x & \text { (2) } \frac{\sin 3 x+\sin x}{2}=\sin 2 x \cos x \\ \sin 4 \theta-\sin 2 \theta \\ \text { 3) } 4 \theta+\cos 2 \theta & \tan \theta\end{array} \)

Let's solve the given equations step by step. ### Equation (1): \[ \cos 5x - \cos 15x = 2 \sin(-10x) \sin 5x \] Using the identity for the difference of cosines: \[ \cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \] we can rewrite the left side: - Let \( A = 15x \) and \( B = 5x \): \[ \cos 5x - \cos 15x = -2 \sin\left(\frac{15x + 5x}{2}\right) \sin\left(\frac{15x - 5x}{2}\right) = -2 \sin(10x) \sin(5x) \] Thus, the equation becomes: \[ -2 \sin(10x) \sin(5x) = 2 \sin(-10x) \sin(5x) \] Since \(\sin(-10x) = -\sin(10x)\), we can rewrite the right side: \[ -2 \sin(10x) \sin(5x) = -2 \sin(10x) \sin(5x) \] This equation holds true for all \(x\) where \(\sin(5x) \neq 0\). ### Equation (2): \[ \frac{\sin 3x + \sin x}{2} = \sin 2x \cos x \] Using the sum-to-product identities: \[ \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \] we can rewrite the left side: - Let \( A = 3x \) and \( B = x \): \[ \frac{\sin 3x + \sin x}{2} = \sin(2x) \cos(x) \] Thus, the equation becomes: \[ \sin(2x) \cos(x) = \sin(2x) \cos(x) \] This equation also holds true for all \(x\) where \(\sin(2x) \neq 0\). ### Equation (3): \[ \sin 4\theta - \sin 2\theta = 4\theta + \cos 2\theta \] Using the double angle identity for sine: \[ \sin 4\theta = 2 \sin 2\theta \cos 2\theta \] we can rewrite the left side: \[ 2 \sin 2\theta \cos 2\theta - \sin 2\theta = 4\theta + \cos 2\theta \] Factoring out \(\sin 2\theta\): \[ \sin 2\theta (2 \cos 2\theta - 1) = 4\theta + \cos 2\theta \] This equation is more complex and may require numerical or graphical methods to solve for specific values of \(\theta\). ### Summary: 1. The first two equations hold true for all \(x\) where \(\sin(5x) \neq 0\) and \(\sin(2x) \neq 0\). 2. The third equation requires further analysis or numerical methods to find specific solutions for \(\theta\). If you need further assistance with the third equation or any specific values, please let me know!