Plane trigonometry, by S.L. Loney.

[Exs. XXVII.] SIDES AND ANGLES OF A TRIANGLE. 187
28. If C=60~, then prove that
1 1 3
a+c b+-c a+b+c
29. In any triangle ABC if D be any point of the base BC, such that
BD: DC:: m: n, prove that
(m +n) cot ADC=ncot B- m cot C,
and (m +?n)2 AD2 = (m + in) (nb)2 + nc2) - nlna2.
30. If in a triangle the bisector of the side c be perpendicular to the
side b, prove that
2tanA+tan C= 0.
31. In any triangle prove that, if 0 be any angle, then
b cos0 =c cos (A - 0) + a cos (C+ 0).
32. If p and q be the perpendiculars from the angular points A and B
on any line passing through the vertex C of the triangle ABC, then
prove that
a2p2 + b2q2 - 2abpq cos C= a2b2 sin2 C.
33. In the triangle ABC, lines OA, OB, and OC are drawn so that
the angles OAB, OBC, and OCA are each equal to w; prove that
cot w= cot A +cot B + cot C,
and cosec' w = cosec2 A + cosec B + cosec' C.