Addition and Subtraction Formulae

(i) sin (A +B) = sin A cos B +cos A sin B
(ii) cos (A +B) = cos A cos B -sin A sin B
(iii) tan (A +B) = (tan A + tan B)/(1 - tan A tan B)
(iv) sin (A -B) = sin A cos B - cos A sin B
(v) cos (A -B) = cos A cos B + sin A sin B
(vi) tan (A -B) = (tan A - tan B)/(1 + tan A tan B)

Corollaries

(i) tan(/4 +A) = tan(45° + A) = (1 + tan A)/(1 -tan A)
(ii) tan(/4 -A) = tan(45° -A) = (1 - tan A)/(1 + tan A)
(iii) cot (A +B) = (cot A cot B - 1)/(cot B + cot A)
(iv) cot (A -B) = (cot A cot B + 1)/(cot B -cot A)
(v) tan (A +B +C) = (tan A +tan B +tan C -tan A tan B tan C)/(1 - tan A tan B - tan B tan C - tan C tan A)
(vi) sin (A +B) sin (A -B) = sin²A -sin²B
(vii) cos (A +B) cos (A -B) = cos²A -sin²B

Illustrative Examples

Example

Using t-ratios of 45° and 60°, evaluate
(i) sin 105°   (ii) tan (13 /12).

Solution

1. sin 105° = sin (60° +45°) = sin 60° cos 45° + cos 60° sin 45°
   =((3)/2)(1/2) + (1/2)(1/2) = (3 + 1)/22
2. 
      =
   = (-1 + 3)(1 - (-1)3) = (3 - 1)/(3 + 1)

Example

1. Prove that cos - sin = 2 cos( + /4)
2. Find the maximum and minimum values of 7 cos + 24 sin .

Solution

1. cos - sin = 2 [(cos ) (1/2) - (sin )(1/2)]=
      2(cos cos/4 -sin sin /4)= 2 cos ( + /4)
2. Since 7² +24² = 49 +576 = 625 = 25²,
        7 cos + 24 sin = 25 [(cos )(7/25) + (sin )(24/25)]
   Now, we can find an angle such that cos = 7/25 and sin = 24/25
   7 cos + 24 sin = 25 (cos cos + sin sin ) = 25 cos (-)
   Now -1 cos (-) 1
   => -25 25 cos (-) 25
   => -25 (7 cos + 24 sin ) 25
   Hence the given expression varies between -25 and 25.
   Therefore, the maximum and minimum values of 7 cos + 24 sin are 25 and -25 respectively.

In general, a cos + b sin = (a² +b²). cos (-), where is given by
cos = a/(a² +b²), sin = b/((a² +b²)
Thus a cos +b sin varies between -(a² +b²) and (a² +b²).

Exercise

1. Prove that sin 75° = [6 + 2]/4
2. Find tan 15° and hence show that tan 15° +cot 15° = 4
3. Evaluate (i) cos 195°   (ii) sin(3 /4)
4. Simplify by reducing to a single term:
   (i) sin 38° cos 22° +cos 38° sin 22°
   (ii) cos 70° cos 10° +sin 70° sin 10°
   (iii) sin (x -y) cos x -cos (x -y) sin x
   (iv) (tan 69° +tan 66°)/(1 -tan 69° tan 66°)
5. Evaluate
   (i) cos 105° +sin 105°  (ii) cos 15° -sin 15°
   (iii) cot 105° -tan 105°
6. (i) A positive acute angle is divided into two parts whose tangents are 1/2 and 1/3. Show that the angle is /4.
   (ii) Prove that tan 22° +tan 23° +tan 22° tan 23° = 1.
   (iii) If A +B = 45°, show that (1 +tan A)(1 +tan B) = 2 and hence find the value of tan 22½°.
   (iv) If A +B = 225°, prove that tan A +tan B = 1 -tan A tan B.
7. Prove that
   (i) tan 70° = tan 20° + 2 tan 50°
   (ii) tan 13A = tan 4A + tan 9A + tan 4A tan 9A tan 13A
   (iii) (1 + tan A)(1 + tan B) = 2 tan A, if A -B = 45°
   (iv) tan (x - y) + tan (y - z) + tan (z - x) = tan (x -y) tan (y - z) tan (z - x)
   (v) tan 56° = (cos 11° +sin 11°)/(cos 11° -sin 11°)
8. Prove that
   sin (x + y) sin (x - y) + sin (y + z) sin (y - z) + sin (z + x) sin (z - x) = 0.
9. Prove that sin (A + B + C) = cos A cos B cos C (tan A + tan B + tan C-tan A tan B tan C).
10. Prove that
    sin A sin (B - C) + sin B sin (C - A) + sin C sin (A -B) = 0.
11. In any quadrilateral ABCD, show that
      cos A cos B -cos C cos D = sin A sin B -sin C sin D.

Answers