For a triangle we know #A+B+C=180^circ.# Supplementary angles have the same sines and opposite cosines. We apply the sum angle formulas and grind it out, simplifying with #cos^2 theta+sin^2 theta=1.#
# sin ^2A + sin ^2 B +sin^ 2 C − 2 cos A cos B cos C #
# = sin ^2A + sin ^2 B +sin^ 2(180^circ - (A+B))− 2 cos A cos B cos (180^circ - (A+B)) #
# = sin ^2A + sin ^2 B +sin^ 2(A+B)+ 2 cos A cos B cos (A+B) #
# = sin ^2A + sin ^2 B +( sin Acos B + cos A sin B )^2+ 2 cos A cos B ( cos A cos B - sin A sin B) #
# = sin ^2A + sin ^2 B + sin^2 Acos^2 B + 2 cos A sin A cos B sin B + cos^2 A sin^2 B + 2 cos^2 A cos^2 B - 2 cos A sin A cos B sin B #
# = sin ^2A + sin ^2 B + sin^2 Acos^2 B + cos^2 A sin^2 B + 2 cos^2 A cos^2 B #
# = ( sin ^2 A + cos^2 A cos^2 B + cos^2 A sin^2 B ) + (sin^2 B + sin^2 A cos^2B + cos ^2 A cos^2 B )#
# = ( sin ^2 A + cos^2 A( cos^2 B + sin^2 B ) ) + (sin^2 B + cos^2 B(sin^2 A + cos ^2 A ) )#
# = ( sin ^2 A + cos^2 A ) + (sin^2 B + cos^2 B )#
# = 1 + 1 #
#= 2#