Proving of trigonometric summation

Good Day,

I need help to prove the following

sin x + sin 2x + sin 3x + ... + sin nx = {sin (n/2)x sin [(n+1)/2]x}/ sin (x/2)

Can it be proved without using induction?

Quote:

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Originally Posted by dd86

Good Day,

I need help to prove the following

sin x + sin 2x + sin 3x + ... + sin nx = {sin (n/2)x sin [(n+1)/2]x}/ sin (x/2)

Can it be proved without using induction?

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Why don't you want to try induction?

I was just wondering if it could be done without using induction, that's why...

Good Day,

I've tried many things to help prove the above summation. However, I don't seem to be getting anywhere.

I've attached a copy of my most logical steps to prove the above summation. Whatever I've done from that point onwards i.e. applying addition and product to sum or sum to product formulas were not included as they were only rough workings.

Attachment 24353

I really hope some one can point me in the right direction to help me prove this.