Hence, in this case,

a = BC= BD - CD=c cos B-(-b cos C),

172. Ex. From the formulae of Art. 164 deduce those of Art. 170 and vice versâ.

The first and third formulae of Art. 164 give

Similarly, the other formulae of Art. 170 may be obtained.
Again, the three formulae of Art. 170 give

a2 + b2 — c2 = a (b cos C + c cos B) +b (c cos A + a cos C) − c (a cos B + b cos A) =2ab cos C.

Similarly, the other formulae of Art. 162 may be found.

173. The student will often meet with identities, which he is required to prove, which involve both the sides and the angles of a triangle.

It is, in general, desirable in the identity to substitute for the sides in terms of the angles, or to substitute for the ratios of the angles in terms of the sides.

Ex. 2. In a triangle prove that

(b2 — c2) cot A + (c2 − a2) cot B + (a2 — b2) cot C = 0.

1

2abck

+ (a2 = b2) a2 + 12 - c2 ]

b2

2abc

[b4c4a2 (b2 - c2)+c4-a4-b2 (c2 - a2)+a+b+ - c2 (a2 - b2)]

c2+a2 - b2

This identity may also be proved by substituting for the sides.
We have, by Art. 163,

Ex. 4. If the sides of a triangle be in Arithmetical Progression, prove that so also are the cotangents of half the angles.

i.e, if a+c=2b, which is relation (1).

Hence if relation (1) be true, so also is relation (2).

6. a2+b2+c2= 2 (bc cos A + ca cos B+ ab cos C).

7. (a2 - b2+c2) tan B = (a2 + b2 — c2) tan C.

12. a2 (cos2 B – cos2 C) + b2 (cos2 C − cos2 A)+c2 (cos2 A — cos2 B)=0.

15. a3 cos (B- C) + b3 cos (C – A) + c3 cos (A – B)=3 abc.

16. In a triangle whose sides are 3, 4, and √38 feet respectively, prove that the largest angle is greater than 120°.