Recall the angle sum identities:
cos(a + b) = cos(a) cos(b) - sin(a) sin(b)
cos(a - b) = cos(a) cos(b) + sin(a) sin(b)
sin(a + b) = sin(a) cos(b) + sin(b) cos(a)
sin(a - b) = sin(a) cos(b) - sin(b) cos(a)
Notice that adding the first two together, and subtract the last from the third, we get two more identities:
cos(a + b) + cos(a - b) = 2 cos(a) cos(b)
sin(a + b) + sin(a - b) = 2 sin(b) cos(a)
Let a = 4x and b = x. Then
cos(5x) + cos(3x) = 2 cos(4x) cos(x)
sin(5x) - sin(3x) = 2 sin(x) cos(4x)
Now,
as required.
