Question

How do you simplify `sin(x)+sin(x-pi)+sin(x+pi)`?

Answer 1

graph{sinx [-10, 10, -5, 5]}

As you can see from the graph, the sine function changes sign at every interval `pi`
If you have a value of say `sinx=0.5` at `x` then `pi` radians further (or back) it will be `-0.5` and vice versa.
(the graph doesn't give the `pi`-values, but they're spaced approximately 3.14 apart).

So `sin(x-pi)=-sinx`
and `sin(x+pi)=-sinx`

Answer
`sinx+sin(x-pi)+sin(x+pi)`
`=sinx-sinx-sinx`
`=-sinx`