Find an identity to solve the equation on the interval Sin 4

Find an identity to solve the equation on the interval Sin 4 x Cos x - Sin x Cos 4x = - root of 2/2.

Solution

use idedntity sinAcosB - sinBcosA = sin(A-B)

given sin4xcosx - sinxcos4x = -sqrt2/2 = -1/sqrt2

sin (4x - x) = sin(3pi/4)

sin3x = sin(3pi/4)

3x = 3pi//4

x = pi/4 = 0.25pi