103 Trigonometry Problems

66 103 Trigonometry Problems
(a)
tan A 2 tan B 2 + tan B 2 tan C 2 + tan C 2 tan A 2 = 1;
(b)
tan A 2 tan B 2 tan C 2 ≤ √
3
9 .
20. Let ABC be an acute-angled triangle. Prove that
(a) tan A + tan B + tan C = tan A tan B tan C;
(b) tan A tan B tan C ≥ 3 √ 3.
21. Let ABC be a triangle. Prove that
cot A cot B + cot B cot C + cot C cot A = 1.
Conversely, prove that if x,y,z are real numbers with xy + yz+ zx = 1, then
there exists a triangle ABC such that cot A = x, cot B = y, and cot C = z.
22. Let ABC be a triangle. Prove that
sin 2 A 2 + B sin2 2 + C sin2 2 + 2 sin A 2 sin B 2 sin C 2 = 1.
Conversely, prove that if x,y,z are positive real numbers such that
x 2 + y 2 + z 2 + 2xyz = 1,
then there is a triangle ABC such that x = sin A 2 , y = sin B 2 , and z = sin C 2 .
23. Let ABC be a triangle. Prove that
(a) sin A 2 sin B 2 sin C 2 ≤ 1 8 ;
(b) sin 2 A 2 + sin2 B 2 + sin2 C 2 ≥ 3 4 ;
(c) cos 2 A 2 + cos2 B 2 + cos2 C 2 ≤ 9 4 ;
(d) cos A 2 cos B 2 cos C 2 ≤ 3√ 3
8 ;