sin x + sin 3x +sin 5x +sin7x+.....+sin 29x what will be its value how to solve it please make me know

S =sin x + sin 3x +sin 5x +sin7x+.....+sin 29x

(2.sin x). S = (2.sin x). (sin x + sin 3x +sin 5x +sin7x+.....+sin 29x)

(2.sin x). S = 2.sin x.sen x +2. without 3x.sin x + 2.sin 5x.sin x + 2.sin7x.sin x + ..... + 2.sin 29x.sin x

2.sinA.sinB = cos (A-B) -cos (A+B), similarly

2.sin x.sin x = cos (x-x) -cos (x + x) = cos (0) -cos (2x) = 1-cos (2x)

2. sin 3x.sin x = cos (3x-x) -cos (3x + x) = cos (2x) -cos (4x)

2.sin 5x.sin x = cos (5x-x) -cos (5x + x) = cos (4x) -cos (6x)

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2.sin 29x.sin x = cos (29x-x) -cos (29x + x) = cos (28x) -cos (30x)

S = 1-cos (2x) + cos (2x) -cos (4x) + cos (4x) -cos (6x) + ... + cos(28x) -cos (30x)

(2.sin x) .S = 1-cos (30x)

(2.sin x) .S = 1- [cos² (15x) -sin (15x)]

(2.sin x) .S = 1-cos² (15x) + sin² (15x)

(2.sin x) .S = sin² (15x) + sin² (15x)

(2.sin x). S = 2.sin² (15x)

S = sin² (15x) / sin (x)