Correct Answer - Option 2 : 4 cos A cos B cos C
Concept:
sin a + sin b = 2 sin (\(\rm a+b\over 2\)) cos (\(\rm a-b\over 2\))
cos a + cos b = 2 cos (\(\rm a+b\over 2\)) cos (\(\rm a-b\over 2\))
sin 2a = 2 sin a cos a
Calculation:
S = sin 2A + sin 2B + sin 2C
S = [2 sin (\(\rm 2A+2B\over 2\)) cos (\(\rm 2A-2B\over 2\))] + 2 sin C cos C
S = 2[sin (A + B) cos (A - B) + sin C cos C]
Given: A + B + C = 90°
So, A + B = 90 - C and C = 90 - (A + B)
S = 2 [sin (90 - C) cos (A - B) + sin (90 - (A + B) cos C]
S = 2 [cos C cos (A - B) + cos (A + B) cos C]
S = 2 cos C [cos (A - B) + cos (A + B)]
S = 2 cos C [2 cos (\(\rm A+B+A-B\over 2\)) cos (\(\rm A+B-(A-B)\over 2\))]
S = 4 cos A cos B cos C