Converting Sum or Difference into Product in Trigonometry:
We have, sin (A + B) = sin A cos B + cos A sin B ………………(i)
sin (A – B) = sin A cos B – cos A sin B …………..(ii)
Adding, we get sin (A + B) + sin (A – B) = 2 sin A cos B
Also subtracting (ii) from (i), we get-
sin (A + B) – sin (A – B) = 2 cos A sin B
Again, cos (A + B) = cos A cos B – sin A sin B ……………(iii)
cos (A – B) = cos A cos B + sin A sin B ………….(iv)
Adding, we get cos (A + B) + cos (A – B) = 2 cos A cos B
Also, subtracting (iii) from (iv), we get-
cos (A – B) – cos (A + B) = 2 sin A sin B
Thus, we get the product formulae in terms of sum and difference of sines or cosines as-
(1) 2 sin A cos B = sin (A + B) + sin (A – B)
(2) 2 cos A sin B = sin (A + B) – sin (A – B)
(3) 2 cos A cos B = cos (A + B) + cos (A – B)
(4) 2 sin A sin B = cos (A- B) – cos (A + B)
Let us now take A + B = C and A – B = D. Then, A = (C + D)/2 and B = (C – D)/2. Therefore, from (1), (2), (3) and (4), we get the sum and differences formulae for sines or cosines in terms of their products as,
(5) sin C + sin D = 2 sin (C + D)/2 . cos (C – D)/2
(6) sin C – sin D = 2 cos (C + D)/2 . sin (C – D)/2
(7) cos C + cos D = 2 cos (C + D)/2 . cos (C – D)/2
(8) cos C – cos D = 2 sin (C + D)/2 . sin (D – C)/2 = -2 sin (C + D)/2 . sin (C – D)/2
