Question

If A + B + C= n, show that
a) Cos A. Sin A + Cos B. Sin B + Cos C. Sin C=2 Sin A. Sin B. Sin C
b) Sin A. Sin B. Cos C+ Sin A. Cos B. Sin C + Cos A. Sin B. Sin C= 1+ Cos A. Cos B. Cos C​

Answer 1

Answer:

Consider the two unit vectors on S2 with spherical polar coordinates (a,0) and (b,π2−c), its components in R3 are:

and (sina,0,cosa)(sinbcos(π2−c),sinbsin(π2−c),cosb)=(sinbsinc,sinbcosc,cosb)

The angle θ between them satisfies:

cosθ=cosacosb+sinasinbsinc

Geometrically, cosθ=1 if and only if these two unit vector coincides. This in turn implies a=b and c=π2. By sine law, the two sides opposites to angle a, b have equal length. So the triangle is an right angled isosceles triangle and sinc=1.

Alternatively, one can use the identity