Calculate the following trigonometric expression: E = cos x+2sin(x/2 +pi/12)sin(x/2 -pi/12)

We notice that pi/12 is half angle of pi/6 = 30
degrees.

Then pi/12 = 15
degrees.

We'll use the
identity:

sin a*sin b = (1/2)*[cos (a-b) - cos
(a+b)]

sin(x/2 +pi/12)sin(x/2 -pi/12) = sin(x/2 +
15)sin(x/2 - 15)

2sin(x/2 + 15)sin(x/2 - 15) = cos(x/2 + 15
- x/2 + 15) - cos(x/2 + 15 + x/2 - 15)

2sin(x/2 +
15)sin(x/2 - 15) = cos(30) - cos(x) (1)

The product term
will be substituted by (1):

E = cos x+ cos 30 - cos
x

E = cos 30

E =
sqrt3/2

The value of the given expression is
E = sqrt3/2.