Given a line with slope y/x, find a simple formula for the slope of a second line that forms a 45 degree angle with this line (find slopes for both 45 degrees more and 45 degrees less.) This can be done without trigonometry.
Find a general formula for any angle.
This requires trigonometry.
Consider the line through the origin and (x,y). Rotate it 90 degrees clockwise to (y,-x). The midpoint is ((x+y)/2,(y-x)/2) so the slope through the origin is
(y-x,x+y) at an angle 45 degrees less than the original.
Similar reasoning (or finding the perpendicular to this) give the slope of 45 degrees more as (x+y/x-y)
For any angle a the formula for a decrease is ((y*cot(a)-x)/(x*cot(a)+y)
and for and increase ((y*cot(a)+x)/(x*cot(a)-y)
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