is there a method of writing sin^n(x) as a sum of sin(mx) [n,m are integer]

[Al-Layth]

I vaguely recall seeing an identity that showed sin^n(x) being expressed in terms of sums sin(mx).
basically writing sin^n(x) as a sum of multiple linear sines but with different factors attached to the argument
I saw this during a complex algebra lesson if that helps

 

[BigBeachBanana]

 

[blamocur]

I vaguely recall seeing an identity that showed sin^n(x) being expressed in terms of sums sin(mx).
basically writing sin^n(x) as a sum of multiple linear sines but with different factors attached to the argument
I saw this during a complex algebra lesson if that helps

Also, when [imath]n[/imath] is odd [imath]\sin n\theta[/imath] can be expressed as sum of powers of [imath]\sin\theta[/imath] (from https://mathworld.wolfram.com/Multiple-AngleFormulas.html):

 

[Al-Layth]

Also, when [imath]n[/imath] is odd [imath]\sin n\theta[/imath] can be expressed as sum of powers of [imath]\sin\theta[/imath] (from https://mathworld.wolfram.com/Multiple-AngleFormulas.html):

View attachment 35880

thanks for the fish