Another interesting thing, if you consider the unit circle, it has a radius of 1, and the x-coordinate of any point on that circle is cos(x) and the y-coordinate is the sin(x) of that point.
PICTURE:
in this picture,
and
now we are trying to find when
, let us call
a point on the unit-circle.
Now, for sin(x) + cos(x) to equal 1, then the point is
because the x-coordinate of P represents
and the y-coordinate represents
Vector OP is the line from the origin to our point P and OP has length 1 (as this is the unit circle), so:
now, dotting this vector with itself, we get:
And since we know
, then:
Expanding & simplifying & canceling, we get:
therefore:
so
will be true when our point is:
(which represent when cos(x) = 0 and sin(x) = 1 AND cos(x) = 1 and sin(x) = 0, respectively)
so these simplify to the two above equations which i gave