Converting Product into Sum/Difference and vice versa

A, B formulae

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)

C, D formulae

1. sin C +sin D = 2 sin (C +D)/2 cos (C -D)/2
2. sin C -sin D = 2 cos (C +D)/2 cos (C -D)/2
3. cos C +cos D = 2 cos (C +D)/2 cos (C -D)/2
4. cos C -cos D = - 2 sin (C +D)/2 sin (C -D)/2 = 2 sin (C+D)/2 sin (D - C)/2

Illustrative Examples

Example

Show that cos 10° +cos 110° +cos 130° = 0

Solution

cos 10° +cos 110° +cos 130° = (cos 110° +cos 10°) +cos 130°
= 2 cos (110° +10°)/2 cos (110° -10°)/2 +cos (180° -50°)
= 2 cos 60° cos 50° -cos 50° = 2.(1/2) cos 50° -cos 50° = 0

Example

Prove that cos 20° cos 40° cos 80° = 1/8

Solution

cos 20° cos 40° cos 80° = (1/2) cos 40° (2 cos 80° cos 20°)
= (1/2) cos 40° (cos (80° +20°) + cos (80° -20°))
= (1/2) cos 40° (cos 100° + cos 60°) = 1/2 cos 40° (cos 100° +1/2)
= (1/4) cos 40° +(1/4) (2 cos 100° cos 40°)
= (1/4) cos 40° + (1/4) (cos 140° +cos 60°)
= (1/4) cos 40° - (1/4)cos 40° +(1/4)(1/2)
                    (Since cos 140° = cos (180° -40°) = -cos 40°)
= 1/8

Exercise

1. Convert the following products into sums or differences
   (i) 2 sin 3 cos 2
   (ii) 2 cos 3 sin 2
   (iii) 2 sin 4 sin 2
   (iv) 2 cos 7 cos 3
2. Evaluate (i) 2 cos 45° cos 15° (ii) 2 sin 75° sin 45°
3. Express each of the following as the product of sines and cosines:
   (i) sin 10 + sin 6
   (ii) sin 10 - sin 6
   (iii) cos 10 + cos 6
   (iv) cos 10 - cos 6
   (v) cos 25° - cos 37°
   (vi) sin 36° + cos 36°
   (vii) sin 80° - cos 70°
4. Prove that
   (i) cos 52° + cos 68° + cos 172° = 0
   (ii) cos 20° + cos 100° + cos 140° = 0
5. Prove that
   (i) cos A +cos (120° -A) +cos (120° +A) = 0
   (ii) cos /8 +cos 3/8 +cos 5/8 +cos 7/8 = 0
   (iii) cos 2 cos /2 -cos 3 cos 9/2 = sin 5 sin 5/2
6. Prove that
   (i) sin 10° sin 50° sin 70° = 1/8
   (ii) sin 10° sin 30° sin 50° sin 70° = 1/16
   (iii) sin 20° sin 40° sin 60° sin 80° = 3/16
7. Prove that
   (i) cos 20° cos 40° cos 60° cos 80° = 1/16
   (ii) cos 10° cos 30° cos 50° cos 70° = 3/16
8. Prove that
   (i) tan 20° tan 40° tan 80° = tan 60°
   (ii) sin 12° sin 48° sin 54° = 1/8
9. Prove that
   (i) sin + sin + sin -sin ( + + ) = 4 sin ( + )/2 sin( + )/2 sin ( + )/2
   (ii) sin + sin +sin + cos ( + + )= 4 cos ( + )/2 cos ( + )/2 cos ( + )/2
10. Prove that
    (i) sin (B -C) cos (A -D) +sin (C -A) cos (B -D) + sin (A -B) cos (C -D) = 0
    (ii) sin (B +C -A) +sin (C +A -B) +sin (A +B -C) - sin (A +B +C) = 4 sin A sin B sin C
11. If sin x +sin y = a and cos x +cos y = b, find the values of
    (i) tan (x +y)/2
    (ii) tan (x -y)/2

Answers