Show that if x, y, and z are consecutive terms of an arithmetic sequence, and tan y is defined, then (sin x + sin y + sin z) / (cos x + cos y + cos z) = tan y.

Question

So I tried letting x = y-k (since x,y,z are consecutive terms of an arithmetic sequence), then z= y+k

So we get (sin(y-k) + siny + sin(y+k)) / (cos(y-k) + cos(y) + cos(y+k)) = tan y+0k

What do I do next?

Answer 1

numerator:
sinycosk-cosysink+siny+sinycosk+cosysink
sinycosk+siny+sinycosk
2sinycosk+siny
siny(2cosk+1)

denominator:
cosycosk+sinysink+cosy+cosycosk-sinysink
2cosycosk+cosy
cosy(2cosk+1)

so,

siny(2cosk+1)
-------------- = tany
cosy(2cosk+1)

Answer 2

THAT HELPED ALOT THANKS