103 Trigonometry Problems

5. Solutions to Advanced Problems 173
with 0 ◦ < x,y,z,w < 90 ◦ . Then |AI| =sec x, |BI|=sec y, |CI| =sec z,
|DI| = sec w, |AD| = |AD 1 |+|D 1 D| = tan x + tan w, and |BC| =
|BB 1 |+|B 1 C|=tan y + tan z. Inequality (∗) becomes
(sec x + sec w) 2 + (sec y + sec z) 2 ≤ (tan x + tan y + tan z + tan w) 2 .
Expanding both sides of the above inequality and applying the identity sec 2 x =
1 + tan 2 x gives
or
4 + 2(sec x sec w + sec y sec z)
≤ 2 tan x tan y + 2 tan x tan z + 2 tan x tan w
+ 2 tan y tan z + 2 tan y tan w + 2 tan z tan w,
2 + sec x sec w + sec y sec z
≤ tan x tan w + tan y tan z + (tan x + tan w)(tan y + tan z).
Note that by the addition and subtraction formulas,
Hence,
Similarly,
1 − tan x tan w =
cos x cos w − sin x sin w
cos x cos w
1 − tan x tan w + sec x sec w =
1 − tan y tan z + sec y sec z =
Adding the last two equations gives
=
cos(x + w)
cos x cos w .
1 + cos(x + w)
cos x cos w .
1 + cos(y + z)
.
cos y cos z
2 + sec x sec w + sec y sec z − tan x tan w − tan y tan z
=
It suffices to show that
or
1 + cos(x + w)
cos x cos w
1 + cos(x + w)
cos x cos w
+
1 + cos(y + z)
cos y cos z
1 + cos(y + z)
+ .
cos y cos z
≤ (tan x + tan w)(tan y + tan z),
s + t ≤ (tan x + tan w)(tan y + tan z),
after setting s = 1+cos(x+w)
1+cos(y+z)
cos x cos w
and t =
cos x cos w
. By the addition and subtraction
formulas, we have
tan x + tan w =
sin x cos w + cos x sin w
cos x cos w
=
sin(x + w)
cos x cos w .