Question

Find and write down a proof that the product of the gradients of two perpendicular lines is -1

Use trigonometry.
Let the slope of a line from A to B be m1. Let the slope of a perpendicular line from A to C be m2. The tangent (or slope) of the line between these two lines is:
tan (a - b) = (m1 + m2)/(1 - m1 m2)
a and b are the ANGLES of the two lines, measured from the x axis.
(This is a trigonometric identity)
Now IF m1 m2 = -1, then
tan (a-b) = (m1 - 1/m1)/0 = infinity
therefore the angle between the two lines is 90 degrees.