Question

1/sin8°+1/sin16°+1/sin32°+....+1/sin8192° is ​

Answer 1

1/sin8°+1/sin16°+1/sin32°+....+1/sin8192° is

Step-by-step explanation:

Weierstrass Substitution

tan(x/2)=sin(x)1+cos(x)

we get

1tan(x/2)−1tan(x)=1+cos(x)sin(x)−cos(x)sin(x)=1sin(x)

The rest is the same telescoping series

∑k=0n1sin(2kx)=∑k=0n[1tan(2k−1x)−1tan(2kx)]=1tan(x/2)−1tan(2nx)

The question has x=8∘ and n=10, so we get

∑k=0101sin(2k8∘)=1tan(4∘)−1tan(8192∘)=1tan(4∘)+1tan(88∘)=1tan(4∘)+tan(2∘)=cos(4∘)sin(4∘)+sin(4∘)1+cos(4∘)=cos(4∘)sin(4∘)+1−cos(4∘)sin(4∘)=1sin(4∘)