Proove sin(45+x) + sin(45-x)=2^1/2*cosx.

We'll apply the sine function to the sum and the
difference of angles 45 and x:

sin (a-b) = sin a*cos b -
sin b*cos a

sin (a+b) = sin a*cos b + sin b*cos
a

We'll put a = 45 and b =
x.

sin (45+x) = sin 45*cos x + sin x*cos
45

sin (45+x) = sqrt2*cos x/2 + sqrt2*sin x/2
(1)

sin (45-x) = sin 45*cos x - sin x*cos
45

sin (45-x) = sqrt2*cos x/2 - sqrt2*sin x/2
(2)

Now, we'll substitute (1) and (2) in the given
identity:

 sin(45+x) + sin(45-x) =
sqrt2*cosx

sqrt2*cos x/2 + sqrt2*sin x/2 + sqrt2*cos x/2 -
sqrt2*sin x/2 = sqrt2*cosx

We'll combine and eliminate like
terms:

2*sqrt2*cos x/2 =
sqrt2*cosx

We'll simplify and we'll
get:

sqrt2*cosx =
sqrt2*cosx

The identity sin(45+x) + sin(45-x)
= sqrt2*cosx is true.