Simplify sin2+sin4+sin6+⋯+sin88
I tried using the sum-to-product formulae, but it was messy, and I didn't know what else to do. Could I get a bit of help? Thanks.
Answer
The angles are in arithmetic progression. Use the formula
n−1∑k=0sin(a+kb)=sinnb2sinb2sin(a+(n−1)b2)
See here for two proofs (using trigonometry, or using complex numbers).
In your case, a=b=2 and n=44.
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