Question

$\sin x+\sin 3x+\sin 5x+\sin 7x=4\cos x\cos 2x\sin 4x$

Answer 1

$\begin{array}{c}\sin x+\sin 3x+\sin 5x+\sin 7x\\ \sin x+7x+\sin 3x+\sin 5x\\ \left[\sin \left(A+B\right)\right]\sin C+\sin D=2\sin \frac{C+D}{2}\cos \frac{C-D}{2}\\ Now,\\ 2\sin \left(\frac{x+7x}{2}\right)\cos \left(\frac{x-7x}{2}\right)+2\sin \left(\frac{3x+5x}{2}\right)\cos \left(\frac{3x-5x}{2}\right)\\ 2\sin 4x\cos 3x+2\sin 4x\cos x\\ 2\sin 4x\left(\cos 3x+\cos x\right)\\ 2\sin 4x\left[2\cos \frac{C+D}{2}\cos \frac{C-D}{2}\right]\\ 2\sin 4x\left[2\cos \frac{3x+x}{2}\cos \frac{3x-x}{2}\right]\\ 2\sin 4x\times 2\cos 2x\cos x\\ 4\sin 4x\cos 2x\cos x.\end{array}$