Sinx,sin2x,2cosxsinx find the next two terms

Question

It is under the topic of sequence and series

Answer 1

The question may be ambiguous:

(1) sin(x), sin(2x), 2cos(x)sin(x); or

(2) sin(x), sin²(x), 2cos(x)sin(x)?

In case (1), sin(2x)=2cos(x)sin(x) by trig identity, so 2nd and 3rd terms are essentially identical.

My guess is:

sin(x), sin(2x), 2cos(x)sin(x), sin(3x), sin(2x)cos(x)+cos(2x)sin(x).

There are a number of variations for the 5th term:

a) sin(2x)cos(x)+cos(2x)sin(x)

b) cos(x)sin(2x)+cos(2x)sin(x)

c) sin(2x)cos(x)+sin(x)cos(2x)

d) cos(x)sin(2x)+sin(x)cos(2x)

e) cos(2x)sin(x)+sin(2x)cos(x)

f) cos(2x)sin(x)+cos(x)sin(2x)

g) sin(x)cos(2x)+sin(2x)cos(x)

h) sin(x)cos(2x)+cos(x)sin(2x)

to name a few.