Proof of the four-parts formula
 

Let the four consecutive parts be B, a, C and b. Then side a, between the angles, is the ‘inner side’, b being the ‘other’; angle C, between sides a and b, is the ‘inner angle’ with B the ‘other’. We want to prove that
 

cos a cosC = sin a cot b − sin C cot B.
 

From the cosine formulas (1) ,

..............(1)

we have

cos b = cos c cos a + sin c sin a cos B (2)
cos c = cos a cos b + sin a sin b cosC. (3)
 

Substituting the right-hand side of equation (3) for cos c in equation (2), we obtain
 

cos b(1 − cos² a) = cos a sin a sin b cosC + sin c sin a cos B.
 

Substituting sin2 a for (1 − cos² a) and dividing throughout by sin a sin b, we get
 

cot b sin a = cos a cosC + sin c/sin b cos B. (4)

Using the sine formula (7.7) it is seen that

.
Hence, equation (4) becomes
 

cos a cosC = sin a cot b − sin C cot B.