Prove: cos(3pie/4 + x) + sin(3pie/4 - x) = 0

cos(3pi/4  + x) + sin(2pi/4 - x) =
0

We will use trigonometric identities to
solve.

We know that:

cos(x+y)
= cosx*cosy - sinx*siny

==> cos(3pi/4+ x) =
cos3pi/4*cosx -
sin3pi/4*sinx

                               =
-1/sqrt2*cosx - 1/sqrt2 *
sinx

                               = -(cosx+sinx)/
sqrt2.............(1)

Also, we know
that:

sin(x-y) = sinx*cosy -
cosx*siny

==> sin(3pi/4  -x) = sin3pi/4*cosx -
cos3pi/4*sinx

                             = 1/sqrt2 * cosx
+ 1/sqrt2 * sinx

                            =
(cosx+sinx)/sqrt2................(2)

Now we will add (1)
and (2).

==> cos(3pi/4+x)+sin(3pi/4-x) =
-(cosx+sinx)/sqrt2 + (cosx+sinx)/sqrt2 = 0

==>
cos(3pi/4+x)+sin(3pi/4-x) = 0 .............q.e.d