Simplify the expression \sin4x+\sin8x+...+\sin4nx

Travis Caldwell

Travis Caldwell

Answered question

2022-04-27

Simplify the expression sin4x+sin8x++sin4nx

Answer & Explanation

Kathleen Keller

Kathleen Keller

Beginner2022-04-28Added 20 answers

If we multiply the whole sum by 2sin(2x), by exploiting:
2sin(4mx)sin(2x)=cos((4m2)x)cos((4m+2)x)
we have, through telescopic sums:
2sin(2x)m=1nsin(4mx)=cos(2x)cos((4n+2)x)

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