What is the value of sum of sqrt2/2+sinx=?

To determine the value of the sum, we'll create matching
functions in the given sum.

Since sin pi/4 = (sqrt2)/2,
we'll substitute the value (sqrt2)/2 by the equivalent function of the angle
pi/4.

We'll transform the sum  into a
product.

sin x + sin pi/4 = sin x +
(sqrt2)/2

sin x + sin pi/4  =  2sin [(x+pi/4)/2]*cos[
(x-pi/4)/2]

sin x + sin pi/4  = 2 sin [(x/2 + pi/8)]*cos[
(x/2 - pi/8)]

We'll use the half angle
identity:

sin [(x+pi/4)/2] = sqrt[2-(sqrt2)*(cos x-sin
x)]/2

cos[ (x-pi/4)/2] = sqrt[2+(sqrt2)*(cos x+sin
x)]/2

sin x + sin pi/4  =
sqrt{[2-(sqrt2)*(cos x-sin x)]*[2+(sqrt2)*(cos x+sin
x)]}/2