43. We have supposed, in the construction of the preceding Article,
that the sides which contain the angle are less than quadrants, for
we have assumed that the tangents at meet and respectively
produced. We must now shew that the formula obtained is true when
these sides are not less than quadrants. This we shall do by special
examination of the cases in which one side or each side is greater than
a quadrant or equal to a quadrant.
(1) Suppose only one of the sides which contain the angle to be
greater than a quadrant, for example . Produce and to
meet at ; and put , .
Then we have from the triangle , by what has already been
proved,
but , , ; thus
(2) Suppose both the sides which contain the angle to be greater
than quadrants. Produce and to meet at ; put ,
; then from the triangle , as before,