Calculate the sum of sin(45-x)+sin(45+x). the result is 0?

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 sum to be
calculated:

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

We'll combine
and eliminate like terms:

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

We'll simplify and we'll
get:

 sin(45+x) + sin(45-x) =
sqrt2*cosx

We can notice that the result of
the sum is not zero, but sqrt2*cosx.