prove this identity: sin(3x) + sin(x) = 2sin(2x)cos(x)

[hndalama]

prove this identity: sin(3x) + sin(x) = 2sin(2x)cos(x)

I started from the left:
sin(2x + x) + sinx
sin2xcosx + cos2xsinx + sinx

I don't know what to do from here.

 

[Ishuda]

I started from the left:
sin(2x + x) + sinx
sin2xcosx + cos2xsinx + sinx

I don't know what to do from here.

We are at the point
g(x) = sin(3x) + sin(x) = sin(2x) cos(x) + cos(2x) sin(x) + sin(x)
We might note that this gives
g(x) = sin(2x) cos(x) + [ cos(2x) + 1 ] sin(x)
= sin(2x) cos(x) + [cos(2x) + cos²(x) + sin²(x)] sin(x)
and consider the cos(2x) term as a step on our way to the final solution.

 

[Deleted member 4993]

I started from the left:
sin(2x + x) + sinx
sin2xcosx + cos2xsinx + sinx

I don't know what to do from here.

sin(3x) + sin(x)

= sin(2x + x) + sin(2x - x)

= [sin(2x)*cos(x) + sin(x)*cos(2x)] + [sin(2x)*cos(x) - sin(x)*cos(2x)]

= continue......

 

[hndalama]

sin(3x) + sin(x)

= sin(2x + x) + sin(2x - x)

= [sin(2x)*cos(x) + sin(x)*cos(2x)] + [sin(2x)*cos(x) - sin(x)*cos(2x)]

= continue......

Thank you

 

[Steven G]

sin(3x) + sin(x)

= sin(2x + x) + sin(2x - x)

= [sin(2x)*cos(x) + sin(x)*cos(2x)] + [sin(2x)*cos(x) - sin(x)*cos(2x)]

= continue......

That is simply beautiful! Very nicely done-Artful!

 

[Ishuda]

That is simply beautiful! Very nicely done-Artful!

Agree