Question

CosA + cosB + cosC = 1 + 4sinA/2 sinB/2 sinC/2

Answer 1

`\cos(A) + \cos( B ) + \cos(C) \\ = 2 \cos( (A + B)/(2) ) \cos( ( A- B)/(2) ) + 1 - 2 { \sin }^(2) (C)/(2) \\ = 1 + 2\cos( (\pi - C)/(2) ) \cos( (A - B)/(2) ) - 2 { \sin }^(2) (C)/(2) \\ = 1 + 2 \sin (C)/(2) \cos( (A - B)/(2) ) - 2 { \sin }^(2) (C)/(2) \\ = 1 + 2 \sin (C)/(2) [ \cos( (A - B)/(2) ) - \sin (C)/(2)] \\ = 1 + 2 \sin (C)/(2) [ \cos( (A - B)/(2) ) - \cos( (A + B)/(2) )] \\ = 1 + 2 \sin(C)/(2) * 2 \sin( ( (A + B)/(2) +(A - B)/(2) )/(2) ) \sin( ( (A + B)/(2) - (A - B)/(2) )/(2) ) \\ = 1 + 4 \sin (A)/(2) \sin (B)/(2) \sin (C)/(2)`