## SUM TO PRODUCT TRIGONOMETRIC IDENTITES

sin C + sin D = 2 sin (C+D)/2 cos (C-D)/2

sin C - sin D = 2 cos (C+D)/2 sin (C-D)/2

cos C + cos D = 2 cos (C+D)/2 cos (C-D)/2

cos C - cos D = 2 sin (C+D)/2 sin (C-D)/2

Example 1 :

Express

sin 4A + sin 2A

in the form of product.

Solution :

Given expression sin 4A + sin 2A exactly matches with

sin C+ sin D = 2 sin (C+D)/2 cos (C-D)/2

Here C  =  4A and D  =  2A

sin 4A + sin 2A  =  2 sin (4A+2A)/2 cos (4A-2A)/2

=  2 sin (6A/2) cos (2A/2)

=  2 sin 3A cos A

Example 2 :

Express

sin 5A - sin 3A

in the form of product.

Solution :

Given expression sin 5A - sin 3A exactly matches with

sin C - sin D   2 cos (C+D)/2 sin (C-D)/2

Here C  =  5A and D  =  3A

sin 5A - sin 3A  =  2 cos (5A+3A)/2 sin (5A-3A)/2

=  2 sin (8A/2) cos (2A/2)

=  2 sin 4A cos A

Example 3 :

Express

cos 3A + cos 7A

in the form of product.

Solution :

Given expression cos 3A + cos 7A exactly matches with

cos C + cos D  =  2 cos (C+D)/2 cos (C-D)/2

Here C  =  3A and D  =  7A

cos 3A + cos 7A  =  2 cos (3A+7A)/2 cos (3A-7A)/2

=  2 sin (10A/2) cos (-4A/2)

=  2 sin 5A cos(-2A)

=  2 sin 5A cos 2A

Example 4 :

Evaluate

cos 15° - cos 75°

Solution :

cos C - cos D  =  -2 sin (C+D)/2 sin (C- D)/2

cos 15° - cos 75°  =  -2 sin (15°+75°)/2 sin (15°-75°)/2

=  -2 sin (90°)/2 sin (-60°)/2

=  -2 sin 45° sin (-30°)

=  2 (1/√2)(-1/2)

=  -1/√2

Example 5 :

Evaluate

sin 75° + sin 15°

Solution :

sin C + sin D  =  2 sin (C+D)/2 cos (C- D)/2

sin 75° + sin 15°  =  2 sin (75°+15°)/2 cos (75°-15°)/2

=  2 sin (90°)/2 cos (60°)/2

=  2 sin 45° cos (30°)

=  2 (1/√2)(√3/2)

√3/√2

Example 6 :

Prove that

(cos 4t - cos 2t)/(sin 4t + sin 2t)  =  - tant

Solution :

(cos 4t - cos 2t)/(sin 4t + sin 2t)

cos 4t - cos 2t  =  -2 sin (4t+2t)/2 sin (4t-2t)/2

cos 4t - cos 2t  =  -2 sin 3t sin t  ----(1)

sin 4t + sin 2t  =  2 sin (4t+2t)/2 cos (4t-2t)/2

sin 4t + sin 2t  =  2 sin3t cost ---(2)

(1)/(2)

=  -2 sin 3t sin t / 2 sin3t cost

=  -sint/cost

=  - tant

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