$P=\sin x(1+2\cos2x+2\cos4x+2\cos6x)$
$=\sin x+2\sin x\cos2x+2\sin x\cos 4x+2\sin x\cos 6x$
$=\sin x+2.\dfrac{1}{2}(\sin3x-\sin x)+2.\dfrac{1}{2}(\sin5x-\sin3x)+2.\dfrac{1}{2}(\sin7x-\sin5x)$
$=\sin x+\sin3x-\sin x+\sin5x-\sin3x+\sin7x-\sin5x$
$=\sin7x$